<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/c9399b22-093a-404c-9c9a-1c9407e09aa7/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsd">1. A vector <span class="_0 blank"> </span><span class="ff2 fs1">a</span></div><div class="t m0 x2 h3 y2 ff3 fs1 fc0 sc0 ls0">G<span class="ff1 fs0 lsb wsd v1"> can be represented in the <span class="ff2 ws0">magnit<span class="_1 blank"> </span>ude-angle</span> notation (<span class="ff2 ls1">a</span>,</span></div><div class="t m1 x3 h4 y1 ff4 fs2 fc0 sc0 lsb ws1">\u03b8</div><div class="t m0 x4 h2 y1 ff1 fs0 fc0 sc0 lsb wsd">), where</div><div class="t m0 x5 h5 y3 ff1 fs3 fc0 sc0 lsb ws2">2 2</div><div class="c x5 y4 w2 h6"><div class="t m0 x6 h7 y5 ff2 fs3 fc0 sc0 lsb">x</div></div><div class="t m0 x7 h7 y6 ff2 fs3 fc0 sc0 lsb">y</div><div class="t m0 x8 h8 y7 ff2 fs0 fc0 sc0 lsb ws3">a<span class="_2 blank"> </span>a a<span class="_3 blank"></span><span class="ff4 ws4">= +</span></div><div class="t m0 x1 h2 y8 ff1 fs0 fc0 sc0 lsb wsd">is the magnitude and </div><div class="t m0 x9 h9 y9 ff1 fs4 fc0 sc0 lsb">1</div><div class="t m0 xa ha ya ff1 fs5 fc0 sc0 lsb ws5">t a n<span class="_4 blank"> </span><span class="ff2 fs4 v2">y</span></div><div class="c xb yb w3 hb"><div class="t m0 x6 hc yc ff2 fs4 fc0 sc0 lsb">x</div></div><div class="c x6 yd w4 hd"><div class="t m0 xc he ye ff2 fs5 fc0 sc0 lsb">a</div><div class="t m0 xc he yf ff2 fs5 fc0 sc0 lsb">a</div><div class="t m2 xd hf y10 ff4 fs6 fc0 sc0 lsb ws6">\u03b8</div><div class="t m0 xe h10 y11 ff4 fs4 fc0 sc0 ls2">\u2212<span class="ff5 fs5 lsb ws7 v3">§ ·</span></div><div class="t m0 xf h11 y12 ff4 fs5 fc0 sc0 ls3">=<span class="ff5 lsb ws7 v4">¨ ¸</span></div><div class="t m0 x10 h12 y13 ff5 fs5 fc0 sc0 lsb ws7">© ¹</div><div class="t m0 x1 h2 y14 ff1 fs0 fc0 sc0 lsb wsd">is the angle<span class="_0 blank"> </span><span class="ff2 fs7">a</span></div><div class="t m0 x11 h13 y15 ff3 fs7 fc0 sc0 ls4">G<span class="ff1 fs0 lsb wsd v1"> makes with the positive <span class="ff2 ls5">x</span> axis. </span></div><div class="t m0 x1 h14 y16 ff1 fs0 fc0 sc0 lsb wsd">(a) Given <span class="ff2 ws0">A<span class="fs8 ls6 v5">x</span></span><span class="ls7 v0">=<span class="ff4 ls8">\u2212</span><span class="lsb">25.0 m and <span class="ff2 ws0">A<span class="fs8 ls9 v5">y</span></span>= 40.0 m, <span class="_5 blank"> </span><span class="fs9 ws8 v6">2 2</span></span></span></div><div class="t m0 x10 h15 y17 ff1 fsa fc0 sc0 lsb ws9">(<span class="_6 blank"> </span>2 5 . 0<span class="_7 blank"> </span>m)<span class="_8 blank"> </span>(4 0 . 0<span class="_7 blank"> </span>m<span class="_1 blank"> </span>)<span class="_9 blank"> </span>4 7 . 2<span class="_7 blank"> </span>m<span class="_a blank"></span><span class="ff2 lsa">A<span class="ff4 lsb wsa">= \u2212<span class="_b blank"> </span>+<span class="_c blank"> </span>=</span></span></div><div class="t m0 x1 h15 y18 ff1 fsa fc0 sc0 lsb wsd">(b) <span class="_d blank"> </span>Recalling <span class="_d blank"> </span>that <span class="_d blank"> </span>tan </div><div class="t m1 x12 h4 y19 ff4 fs2 fc0 sc0 lsb ws1">\u03b8</div><div class="t m0 x13 h15 y19 ff1 fsa fc0 sc0 lsb wsd"> <span class="_d blank"> </span>= <span class="_d blank"> </span>tan <span class="_d blank"> </span>(</div><div class="t m1 x14 h4 y19 ff4 fs2 fc0 sc0 lsb ws1">\u03b8</div><div class="t m0 x15 h16 y19 ff1 fsa fc0 sc0 lsb wsd"> <span class="_d blank"> </span>+ <span class="_d blank"> </span>180°), <span class="_d blank"> </span>tan<span class="fs8 wsb v6">!1</span> <span class="_d blank"> </span>[(40.0 <span class="_d blank"> </span>m)/ <span class="_d blank"> </span>(! <span class="_d blank"> </span>25.0 <span class="_d blank"> </span>m)] <span class="_d blank"> </span>= <span class="_d blank"> </span>! <span class="_d blank"> </span>58° <span class="_d blank"> </span>or <span class="_e blank"> </span>122°. </div><div class="t m0 x1 h15 y1a ff1 fsa fc0 sc0 lsb wsd">Noting that <span class="_1 blank"> </span>the <span class="_1 blank"> </span>vector is <span class="_1 blank"> </span>in <span class="_1 blank"> </span>the <span class="_1 blank"> </span>third quadrant <span class="_1 blank"> </span>(by <span class="_1 blank"> </span>the signs <span class="_1 blank"> </span>of <span class="_1 blank"> </span>its <span class="ff2 wsc">x</span> <span class="_1 blank"> </span>and <span class="_1 blank"> </span><span class="ff2 wsc">y</span> <span class="_1 blank"> </span>components) we </div><div class="t m0 x1 h15 y1b ff1 fsa fc0 sc0 lsb wsd">see that <span class="_1 blank"> </span>122° is <span class="_1 blank"> </span>the <span class="_1 blank"> </span>correct answer. <span class="_1 blank"> </span>The graphical <span class="_1 blank"> </span>calculator "shortcuts# <span class="_1 blank"> </span>mentioned above </div><div class="t m0 x1 h15 y1c ff1 fsa fc0 sc0 lsb wsd">are designed to correctly choose the right <span class="_f blank"></span>possibility. </div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><div class="t m0 x16 h17 y1d ff1 fsb fc0 sc0 lsb wsd">3. The <span class="ff2 fsa wsc">x</span> and the <span class="ff2 fsa wsc">y</span> components of a vector </div><div class="c x17 y1e w5 h18"><div class="t m0 x6 h19 y1f ff3 fsc fc0 sc0 lsb">G</div></div><div class="t m0 x18 h1a y20 ff2 fsc fc0 sc0 lsc">a<span class="ff1 fsb lsb wsd v0"> lying on the <span class="ff2 fsa wsc">xy</span> plane are given by </span></div><div class="t m0 x19 h1b y21 ff1 fsd fc0 sc0 lsb wse">c o s<span class="_10 blank"> </span>,<span class="_11 blank"> </span>s i n</div><div class="t m0 x1a h1c y22 ff2 fse fc0 sc0 lsb wsf">x y</div><div class="t m0 x1b h1d y23 ff2 fsd fc0 sc0 lsb ws10">a a<span class="_12 blank"> </span>a<span class="_13 blank"> </span>a</div><div class="t m2 xa h1e y24 ff4 fsf fc0 sc0 lsb ws11">\u03b8 \u03b8</div><div class="t m0 x1c h1f y24 ff4 fsd fc0 sc0 lsb ws12">= =</div><div class="t m0 x16 h20 y25 ff1 fsd fc0 sc0 lsb ws13">where <span class="fs10 ws14">| |<span class="_14 blank"></span><span class="ff2 ws15">a a<span class="_15 blank"></span><span class="ff4 lsd">=<span class="ff3 lse v7">G</span><span class="ff1 fsd lsb wsd"> is the magnitude and </span></span></span></span></div><div class="t m2 x1d h1e y25 ff4 fsf fc0 sc0 lsb ws16">\u03b8</div><div class="t m0 x19 h1b y25 ff1 fsd fc0 sc0 lsb wsd"> is the angle between </div><div class="c x1e y26 w5 h21"><div class="t m0 x6 h22 y27 ff3 fs10 fc0 sc0 lsb">G</div></div><div class="t m0 x1f h23 y25 ff2 fs10 fc0 sc0 lsf">a<span class="ff1 fsd lsb wsd"> and the positive <span class="ff2 ls10">x</span> axis. </span></div><div class="t m0 x16 h1b y28 ff1 fsd fc0 sc0 lsb wsd">(a) The <span class="ff2 ls11">x</span> component of </div><div class="c x20 y29 w5 h24"><div class="t m0 x6 h25 y2a ff3 fs11 fc0 sc0 lsb">G</div></div><div class="c x6 yd w4 hd"><div class="t m0 x21 h1b y2b ff2 fs11 fc0 sc0 ls12">a<span class="ff1 fsd lsb wsd"> is given by </span><span class="lsb ws17">a<span class="fs8 wsb v5">x</span><span class="ff1 fsd wsd"> = 7.3 cos 250° = ! 2.5 m. </span></span></div><div class="t m0 x16 h1b y2c ff1 fsd fc0 sc0 lsb wsd">(b) and the <span class="ff2 fs11 ls13">y</span> component is given by <span class="ff2 fs11 ws17">a<span class="fs8 wsb v5">y</span></span> = 7.3 sin 250° = ! 6.9 m. </div><div class="t m0 x16 h1b y2d ff1 fsd fc0 sc0 lsb wsd">In <span class="_1 blank"> </span>considering <span class="_1 blank"> </span>the <span class="_1 blank"> </span>variety <span class="_16 blank"> </span>of <span class="_1 blank"> </span>ways <span class="_1 blank"> </span>to <span class="_1 blank"> </span>compute <span class="_1 blank"> </span>these, <span class="_1 blank"> </span>we <span class="_1 blank"> </span>note <span class="_16 blank"> </span>that <span class="_1 blank"> </span>the <span class="_1 blank"> </span>vector <span class="_1 blank"> </span>is <span class="_1 blank"> </span>70<span class="_1 blank"> </span>° <span class="_1 blank"> </span>below </div><div class="t m0 x16 h1b y2e ff1 fsd fc0 sc0 lsb wsd">the <span class="_16 blank"> </span>! <span class="_16 blank"> </span><span class="ff2 fs11 ls14">x</span> <span class="_d blank"> </span>axis, <span class="_1 blank"> </span>so <span class="_d blank"> </span>the <span class="_1 blank"> </span>components <span class="_d blank"> </span>could <span class="_1 blank"> </span>also <span class="_d blank"> </span>have <span class="_1 blank"> </span>been <span class="_d blank"> </span>found <span class="_1 blank"> </span>from <span class="_d blank"> </span><span class="ff2 fs11 ws17">a<span class="fs8 wsb v5">x</span></span> <span class="_1 blank"> </span>= <span class="_d blank"> </span>! <span class="_16 blank"> </span>7.3 <span class="_16 blank"> </span>cos <span class="_d blank"> </span>70° <span class="_16 blank"> </span>and </div><div class="t m0 x16 h1b y2f ff2 fs11 fc0 sc0 lsb ws17">a<span class="fs8 wsb v5">y</span><span class="ff1 fsd wsd"> <span class="_1 blank"> </span>= <span class="_1 blank"> </span>! <span class="_1 blank"> </span>7.3 <span class="_1 blank"> </span>sin <span class="_1 blank"> </span>70°<span class="_1 blank"> </span>. <span class="_1 blank"> </span>In <span class="_1 blank"> </span>a <span class="_1 blank"> </span>similar <span class="_1 blank"> </span>vein, <span class="_1 blank"> </span>we <span class="_1 blank"> </span>note<span class="_1 blank"> </span> <span class="_1 blank"> </span>that <span class="_1 blank"> </span>the <span class="_1 blank"> </span>ve<span class="_1 blank"> </span>ctor <span class="_1 blank"> </span>is <span class="_1 blank"> </span>20° <span class="_16 blank"> </span>to <span class="_1 blank"> </span>the <span class="_1 blank"> </span>left <span class="_16 blank"> </span>from <span class="_1 blank"> </span>the <span class="_1 blank"> </span>! <span class="_1 blank"> </span></span>y</div><div class="t m0 x16 h1b y30 ff1 fsd fc0 sc0 lsb wsd">axis, so one could use <span class="ff2 fs11 ws17">a<span class="fs8 wsb v5">x</span></span> = ! 7.3 sin 20° and <span class="ff2 fs11 ws17">a<span class="fs8 wsb v5">y</span></span> = ! 7.3 cos 20° <span class="_1 blank"> </span>to achieve the same results. </div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><div class="t m0 x1 h1b y31 ff1 fsd fc0 sc0 lsb wsd">4. (a) The height is <span class="ff2 fs11 ws17">h</span> = <span class="ff2 fs11 ws17">d</span> sin</div><div class="t m2 x22 h1e y31 ff4 fsf fc0 sc0 lsb ws16">\u03b8</div><div class="t m0 x23 h1b y31 ff1 fsd fc0 sc0 lsb wsd">, where <span class="ff2 fs11 ws17">d</span> = 12.5 m and </div><div class="t m2 x24 h1e y31 ff4 fsf fc0 sc0 lsb ws16">\u03b8</div><div class="t m0 x25 h1b y31 ff1 fsd fc0 sc0 lsb wsd"> = 20.0°. Therefore, <span class="ff2 fs11 ws17">h</span> = 4.28 m. </div><div class="t m0 x1 h1b y32 ff1 fsd fc0 sc0 lsb wsd">(b) The horizontal distance is <span class="ff2 fs11 ws17">d</span> cos</div><div class="t m2 x26 h1e y33 ff4 fsf fc0 sc0 lsb ws16">\u03b8</div><div class="t m0 x27 h1b y33 ff1 fsd fc0 sc0 lsb wsd"> = 11.7 m. </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x16 y34 w6 h26" alt="" src="https://files.passeidireto.com/c9399b22-093a-404c-9c9a-1c9407e09aa7/bg4.png"><div class="t m0 x1 h27 y35 ff6 fs12 fc0 sc0 lsb wsd">(a) We compute the d<span class="_1 blank"> </span>istance from one <span class="_17 blank"> </span>corner to t<span class="_1 blank"> </span>he diametrically opposite <span class="_17 blank"> </span>corner: </div><div class="t m0 x28 h28 y36 ff6 fs13 fc0 sc0 lsb ws18">2 2<span class="_18 blank"> </span>2</div><div class="t m0 x29 h27 y37 ff6 fs12 fc0 sc0 lsb wsd">(3.00 m<span class="_1 blank"> </span>)<span class="_8 blank"> </span>(3.70<span class="_1 blank"> </span> m<span class="_1 blank"> </span>)<span class="_8 blank"> </span>(4.30<span class="_16 blank"> </span> m)<span class="_19 blank"></span><span class="ff7 ws19">+ +<span class="_1a blank"> </span><span class="ff6">.</span></span></div><div class="c x6 yd w4 hd"><div class="t m0 x1 h27 y38 ff6 fs12 fc0 sc0 lsb wsd">(b) <span class="_1 blank"> </span>The <span class="_1 blank"> </span>displa<span class="_1 blank"> </span>cement <span class="_1 blank"> </span>vector <span class="_1 blank"> </span>is <span class="_1 blank"> </span>al<span class="_1 blank"> </span>ong <span class="_1 blank"> </span>the <span class="_1 blank"> </span>straight <span class="_16 blank"> </span>line <span class="_1 blank"> </span>from <span class="_1 blank"> </span>the <span class="_1 blank"> </span>beginning <span class="_16 blank"> </span>to <span class="_1 blank"> </span>the <span class="_1 blank"> </span>end <span class="_16 blank"> </span>point </div><div class="t m0 x1 h27 y39 ff6 fs12 fc0 sc0 lsb wsd">of <span class="_1 blank"> </span>the <span class="_1 blank"> </span>trip. <span class="_1 blank"> </span> <span class="_16 blank"> </span>Since <span class="_1 blank"> </span>a <span class="_1 blank"> </span>straight <span class="_1 blank"> </span>lin<span class="_1 blank"> </span>e <span class="_1 blank"> </span>is <span class="_1 blank"> </span>the <span class="_1 blank"> </span>shortest <span class="_1 blank"> </span>distance <span class="_16 blank"> </span>between <span class="_1 blank"> </span>two <span class="_1 blank"> </span>p<span class="_1 blank"> </span>oints, <span class="_1 blank"> </span>the <span class="_1 blank"> </span>lengt<span class="_1 blank"> </span>h <span class="_1 blank"> </span>of </div><div class="t m0 x1 h27 y3a ff6 fs12 fc0 sc0 lsb wsd">the path cannot be less than the magnitude of the displacement. </div><div class="t m0 x1 h27 y3b ff6 fs12 fc0 sc0 lsb wsd">(c) <span class="_d blank"> </span>It <span class="_16 blank"> </span>can <span class="_d blank"> </span>be <span class="_16 blank"> </span>gre<span class="_1 blank"> </span>ater, <span class="_d blank"> </span>however. <span class="_16 blank"> </span>The <span class="_d blank"> </span>fly <span class="_d blank"> </span>m<span class="_f blank"></span>ight, <span class="_d blank"> </span>for <span class="_16 blank"> </span>example, <span class="_d blank"> </span>crawl <span class="_16 blank"> </span>along <span class="_d blank"> </span>the <span class="_d blank"> </span>edges <span class="_16 blank"> </span>of <span class="_d blank"> </span>the </div><div class="t m0 x1 h27 y3c ff6 fs12 fc0 sc0 lsb wsd">room. <span class="_1b blank"> </span>Its <span class="_1b blank"> </span>displacement <span class="_1b blank"> </span>would <span class="_1b blank"> </span>be<span class="_1 blank"> </span> <span class="_1b blank"> </span>the <span class="_1b blank"> </span>same <span class="_1b blank"> </span>but <span class="_6 blank"> </span>the <span class="_1b blank"> </span>path <span class="_1b blank"> </span>length <span class="_6 blank"> </span>would <span class="_1b blank"> </span>be </div><div class="t m0 x2a h29 y3d ff6 fs7 fc0 sc0 lsb wsd">11.0<span class="_1 blank"> </span> m.<span class="_1c blank"></span><span class="ff2 fs14 ws1a">w h<span class="_1d blank"></span><span class="ff7 fs7 ws1b">+<span class="_1e blank"> </span>+ =<span class="_1f blank"></span><span class="ff3 fs14">A</span></span></span></div><div class="t m0 x1 h27 y3e ff6 fs12 fc0 sc0 lsb wsd">(d) T<span class="_1 blank"> </span>he pa<span class="_1 blank"> </span>th l<span class="_1 blank"> </span>ength <span class="_1 blank"> </span>is t<span class="_1 blank"> </span>he same <span class="_1 blank"> </span>as <span class="_1 blank"> </span>the <span class="_1 blank"> </span>magnitude of <span class="_1 blank"> </span>the <span class="_1 blank"> </span>displacement <span class="_1 blank"> </span>if <span class="_1 blank"> </span>the <span class="_1 blank"> </span>fly <span class="_1 blank"> </span>flies <span class="_1 blank"> </span>along </div><div class="t m0 x1 h27 y3f ff6 fs12 fc0 sc0 lsb wsd">the displacement vector. </div><div class="t m0 x1 h27 y40 ff6 fs12 fc0 sc0 lsb wsd">(e) <span class="_1 blank"> </span>We <span class="_1 blank"> </span>take <span class="_16 blank"> </span>the <span class="_1 blank"> </span><span class="ff2 ls15">x</span> <span class="_1 blank"> </span>axis <span class="_16 blank"> </span>to <span class="_1 blank"> </span>be <span class="_1 blank"> </span>out <span class="_16 blank"> </span>of <span class="_1 blank"> </span>the <span class="_1 blank"> </span>page, <span class="_16 blank"> </span>the <span class="_1 blank"> </span><span class="ff2 ls16">y</span> <span class="_1 blank"> </span>axis <span class="_1 blank"> </span>t<span class="_1 blank"> </span>o <span class="_1 blank"> </span>be <span class="_1 blank"> </span>to <span class="_16 blank"> </span>the <span class="_1 blank"> </span>right, <span class="_16 blank"> </span>and <span class="_1 blank"> </span>the <span class="_1 blank"> </span><span class="ff2 ls17">z</span> <span class="_16 blank"> </span>axis <span class="_1 blank"> </span>t<span class="_1 blank"> </span>o </div><div class="t m0 x1 h27 y41 ff6 fs12 fc0 sc0 lsb wsd">be upward. <span class="_1 blank"> </span>Then the <span class="_1 blank"> </span><span class="ff2 ls18">x</span> component of<span class="_1 blank"> </span> the d<span class="_1 blank"> </span>isplacement is <span class="ff2 ws1c">w</span> = <span class="_1 blank"> </span>3.70 m, the <span class="_1 blank"> </span><span class="ff2 ls15">y</span> <span class="_1 blank"> </span>component of </div><div class="t m0 x1 h27 y42 ff6 fs12 fc0 sc0 lsb wsd">the displacement is 4.30 m, and the <span class="ff2 ls19">z</span> component is 3.00 m<span class="_f blank"></span>. Thus, </div><div class="t m0 x2b h27 y43 ff6 fs12 fc0 sc0 lsb ws1d">!<span class="_20 blank"> </span>! !</div><div class="t m0 x2c h27 y44 ff6 fs12 fc0 sc0 lsb wsd">(3.70<span class="_1 blank"> </span> m)<span class="_0 blank"> </span>i<span class="_21 blank"> </span>(<span class="_0 blank"> </span>4.30 m<span class="_f blank"></span>)<span class="_7 blank"> </span>j<span class="_22 blank"> </span>(<span class="_d blank"> </span>3.00 m<span class="_0 blank"> </span>)k<span class="_23 blank"></span><span class="ff2 ls1a">d<span class="ff7 lsb ws1e">=<span class="_24 blank"> </span>+ +</span></span></div><div class="t m0 x2d h2a y45 ff3 fs12 fc0 sc0 lsb">G</div><div class="t m0 x2e h27 y44 ff6 fs12 fc0 sc0 lsb">.</div><div class="t m0 x1 h27 y46 ff6 fs12 fc0 sc0 lsb wsd">An equally correct answer is gotten by interchanging the length, width, and height<span class="_1 blank"> </span>. </div></div><div class="t m0 x1 h27 y47 ff6 fs12 fc0 sc0 lsb wsd">7. The length unit meter is understood throughout the calculation. </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x2f y48 w7 h2b" alt="" src="https://files.passeidireto.com/c9399b22-093a-404c-9c9a-1c9407e09aa7/bg5.png"><div class="t m0 x1 h27 y49 ff6 fs12 fc0 sc0 lsb wsd">8. <span class="_16 blank"> </span>We <span class="_d blank"> </span>label <span class="_16 blank"> </span>the <span class="_d blank"> </span>displacement <span class="_16 blank"> </span>vectors </div><div class="c x30 y4a w5 h2c"><div class="t m0 x6 h2a y4b ff3 fs12 fc0 sc0 lsb">G</div></div><div class="t m0 x31 h27 y49 ff2 fs12 fc0 sc0 ls1b">A<span class="ff6 lsb">,</span></div><div class="c x32 y4a w8 h2c"><div class="t m0 x6 h2a y4b ff3 fs12 fc0 sc0 lsb">G</div></div><div class="c x0 y4c w9 h2d"><div class="t m0 x6 h2e y4d ff2 fs12 fc0 sc0 lsb">B</div></div><div class="t m0 x33 h27 y49 ff6 fs12 fc0 sc0 lsb wsd"> and </div><div class="c x7 y4a w5 h2c"><div class="t m0 x6 h2a y4b ff3 fs12 fc0 sc0 lsb">G</div></div><div class="c xb y4c wa h2d"><div class="t m0 x6 h2e y4d ff2 fs12 fc0 sc0 lsb">C</div></div><div class="t m0 x34 h27 y49 ff6 fs12 fc0 sc0 lsb wsd"> (and <span class="_d blank"> </span>denote <span class="_16 blank"> </span>the <span class="_d blank"> </span>result <span class="_16 blank"> </span>of<span class="_1 blank"> </span> <span class="_d blank"> </span>their <span class="_16 blank"> </span>vector </div><div class="t m0 x1 h27 y4e ff6 fs12 fc0 sc0 lsb wsd">sum <span class="_1 blank"> </span>as </div><div class="c x35 y4f w8 h2f"><div class="t m0 x6 h22 y50 ff3 fs10 fc0 sc0 lsb">G</div></div><div class="c x35 y51 wb h30"><div class="t m0 x6 h23 y52 ff2 fs10 fc0 sc0 lsb">r</div></div><div class="t m0 x36 h31 y53 ff6 fs12 fc0 sc0 lsb wsd">). <span class="_16 blank"> </span>We <span class="_16 blank"> </span>choose <span class="_16 blank"> </span><span class="ff2 ws1c">east</span> <span class="_16 blank"> </span>as <span class="_16 blank"> </span>the <span class="_0 blank"> </span><span class="fs10 v8">!</span></div><div class="t m0 x37 h32 y53 ff6 fs10 fc0 sc0 ls1c">i<span class="fs12 lsb wsd"> direction <span class="_16 blank"> </span>(+<span class="ff2 ls18">x</span> <span class="_16 blank"> </span>directi<span class="_1 blank"> </span>on) <span class="_16 blank"> </span>and <span class="_16 blank"> </span><span class="ff2 ws1c">north</span> <span class="_d blank"> </span>as <span class="_1 blank"> </span>the <span class="_0 blank"> </span><span class="fs15 v8">!</span></span></div><div class="t m0 x38 h33 y54 ff6 fs15 fc0 sc0 ls1d">j<span class="fs12 lsb wsd v0"> direction </span></div><div class="t m0 x1 h34 y55 ff6 fs12 fc0 sc0 lsb ws1c">(+<span class="ff2 ls1e">y</span><span class="fs15 wsd v0"> direction). We n<span class="_f blank"></span>ote that the angle between </span></div><div class="c x39 y56 w8 h35"><div class="t m0 x6 h2a y57 ff3 fs12 fc0 sc0 lsb">G</div></div><div class="c x3a y58 wa h36"><div class="t m0 x6 h2e y59 ff2 fs12 fc0 sc0 lsb">C</div></div><div class="t m0 x3b h33 y5a ff6 fs15 fc0 sc0 lsb wsd"> and the <span class="ff2 fs12 ls16">x</span> axis is 6<span class="_f blank"></span>0°. Thus, </div><div class="c x6 yd w4 hd"><div class="t m3 x1e h37 y5b ff8 fs16 fc0 sc0 lsb ws1f">(<span class="_4 blank"> </span>)<span class="_25 blank"> </span>( )</div><div class="t m0 x3c h33 y5c ff6 fs15 fc0 sc0 lsb">!</div><div class="t m0 x10 h33 y5d ff6 fs15 fc0 sc0 lsb wsd">(50<span class="_1 blank"> </span> km)<span class="_0 blank"> </span>i</div><div class="t m0 x3c h33 y5e ff6 fs15 fc0 sc0 lsb">!</div><div class="t m0 x10 h33 y5f ff6 fs15 fc0 sc0 lsb wsd">(30<span class="_1 blank"> </span> km) j</div><div class="t m0 x3d h33 y60 ff6 fs15 fc0 sc0 lsb ws20">! !</div><div class="t m0 x3a h33 y61 ff6 fs15 fc0 sc0 lsb wsd">(<span class="_1 blank"> </span>25 km<span class="_1 blank"> </span>)<span class="_26 blank"> </span>cos<span class="_27 blank"> </span>60<span class="_28 blank"> </span>i<span class="_29 blank"> </span>+<span class="_26 blank"> </span>(25 km<span class="_0 blank"> </span>)<span class="_d blank"> </span>sin<span class="_2a blank"> </span>60<span class="_2b blank"> </span>j</div><div class="t m0 x3e h2e y62 ff2 fs12 fc0 sc0 lsb">A</div><div class="t m0 x3e h2e y63 ff2 fs12 fc0 sc0 lsb">B</div><div class="t m0 xa h2e y64 ff2 fs12 fc0 sc0 lsb">C</div><div class="t m0 x3f h38 y65 ff8 fs12 fc0 sc0 lsb">=</div><div class="t m0 x3f h38 y66 ff8 fs12 fc0 sc0 lsb">=</div><div class="t m0 x3f h39 y67 ff7 fs12 fc0 sc0 lsb ws21">= °<span class="_2c blank"> </span>°</div><div class="t m0 x0 h2a y68 ff3 fs12 fc0 sc0 lsb">G</div><div class="t m0 x0 h2a y69 ff3 fs12 fc0 sc0 lsb">G</div><div class="t m0 x0 h2a y6a ff3 fs12 fc0 sc0 lsb">G</div><div class="t m0 x1 h33 y6b ff6 fs15 fc0 sc0 lsb wsd">(a) The total displacem<span class="_f blank"></span>ent of the car from<span class="_f blank"></span> its initial position <span class="_f blank"></span>is represented by </div><div class="t m0 x40 h33 y6c ff6 fs15 fc0 sc0 lsb ws22">! !</div><div class="t m0 x41 h33 y6d ff6 fs15 fc0 sc0 lsb wsd">(62.5<span class="_e blank"> </span> km<span class="_1 blank"> </span>)<span class="_0 blank"> </span>i<span class="_2d blank"> </span>(51.7<span class="_2e blank"> </span>km) j</div><div class="t m0 x42 h2e y6e ff2 fs12 fc0 sc0 lsb ws23">r A B C<span class="_2f blank"></span><span class="ff7 fs15 ws24">= +<span class="_8 blank"> </span>+<span class="_30 blank"> </span>=<span class="_31 blank"> </span>+</span></div><div class="t m0 x43 h2a y6f ff3 fs12 fc0 sc0 lsb ws25">G G</div><div class="t m0 x44 h2a y70 ff3 fs12 fc0 sc0 lsb">G</div><div class="t m0 x42 h2a y71 ff3 fs12 fc0 sc0 lsb">G</div><div class="t m0 x1 h33 y72 ff6 fs15 fc0 sc0 lsb wsd">which means th<span class="_f blank"></span>at its magnitud<span class="_f blank"></span>e is </div><div class="t m0 xa h3a y73 ff6 fs17 fc0 sc0 lsb ws26">2 2</div><div class="t m0 x45 h3b y74 ff6 fs18 fc0 sc0 lsb ws27">(62.5<span class="_e blank"> </span>km )<span class="_8 blank"> </span>(51.7<span class="_0 blank"> </span>km)<span class="_32 blank"> </span>81<span class="_7 blank"> </span>km.<span class="_33 blank"></span><span class="ff2 ls1f">r<span class="ff7 lsb ws28">=<span class="_34 blank"> </span>+ =</span></span></div><div class="t m0 x46 h3c y75 ff3 fs18 fc0 sc0 lsb">G</div><div class="t m0 x1 h3d y76 ff6 fs15 fc0 sc0 lsb wsd">(b) The <span class="_1 blank"> </span>angle (counterclockwise from +<span class="ff2 fs12 ls16">x</span> axis) <span class="_1 blank"> </span>is tan<span class="fs8 wsb v6">\u20221</span> <span class="_1 blank"> </span>(51.7 km/62.5 km) <span class="ff7 ws29">=</span> <span class="_1 blank"> </span>40°, <span class="_1 blank"> </span>which is </div><div class="t m0 x1 h33 y77 ff6 fs15 fc0 sc0 lsb wsd">to <span class="_1 blank"> </span>say <span class="_1 blank"> </span>that <span class="_16 blank"> </span>it <span class="_1 blank"> </span>points <span class="_16 blank"> </span>40° <span class="_1 blank"> </span><span class="ff2 fs12 v0">north <span class="_1 blank"> </span>of <span class="_16 blank"> </span>east</span><span class="v0">. <span class="_16 blank"> </span>Although <span class="_1 blank"> </span>the <span class="_1 blank"> </span>resultant </span></div></div><div class="c x47 y78 wc h24"><div class="t m0 x6 h2a y79 ff3 fs12 fc0 sc0 lsb">G</div></div><div class="c x47 y7a wd h3e"><div class="t m0 x6 h2e y7b ff2 fs12 fc0 sc0 lsb">r</div></div><div class="t m0 x48 h33 y7c ff6 fs15 fc0 sc0 lsb wsd"> is <span class="_1 blank"> </span>shown <span class="_16 blank"> </span>in <span class="_1 blank"> </span>our <span class="_16 blank"> </span>sketch, <span class="_1 blank"> </span>it </div><div class="t m0 x1 h33 y7d ff6 fs15 fc0 sc0 lsb wsd">would be a direct line fr<span class="_f blank"></span>om the #tail$ of<span class="_f blank"></span> </div><div class="c x49 y7e wc h35"><div class="t m0 x6 h2a y7f ff3 fs12 fc0 sc0 lsb">G</div></div><div class="t m0 x41 h33 y80 ff2 fs12 fc0 sc0 ls20">A<span class="ff6 fs15 lsb wsd"> to the #head$ of </span></div><div class="c x4a y7e wc h35"><div class="t m0 x6 h2a y7f ff3 fs12 fc0 sc0 lsb">G</div></div><div class="c x4b y81 wa h3f"><div class="t m0 x6 h2e y82 ff2 fs12 fc0 sc0 lsb">C</div></div><div class="t m0 x47 h33 y80 ff6 fs15 fc0 sc0 lsb">.</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x4c y83 we h40" alt="" src="https://files.passeidireto.com/c9399b22-093a-404c-9c9a-1c9407e09aa7/bg6.png"><div class="t m0 x1 h33 y84 ff6 fs15 fc0 sc0 lsb wsd">(c) The angle between the re<span class="_f blank"></span>sultant and the +</div><div class="t m0 xe h33 y85 ff2 fs12 fc0 sc0 ls15">x<span class="ff6 fs15 lsb wsd"> axis is given by</span></div><div class="t m2 x4d h1e y86 ff9 fsf fc0 sc0 lsb wsd">\u03b8 </div><div class="t m0 x4e h41 y86 ff6 fs15 fc0 sc0 lsb wsd">= tan<span class="ff1 fs8 wsb v6">!1</span><span class="ff1 ls21">(<span class="ff2 fs12 ls17">r<span class="fs8 lsb wsb v5">y</span></span><span class="ls22">/<span class="ff2 fs12 ls23">r<span class="fs8 lsb wsb v5">x</span></span><span class="lsb">) = tan<span class="fs8 wsb v6">!1</span> [(10 m)/(<span class="_f blank"></span> !9.0 m)] = <span class="_f blank"></span>! 48° or 132°. </span></span></span></div><div class="t m0 x1 h33 y87 ff1 fs15 fc0 sc0 lsb wsd">Since <span class="_35 blank"> </span>the <span class="_35 blank"> </span><span class="ff2 fs12 ls18 v0">x</span><span class="v0"> <span class="_35 blank"> </span>component <span class="_35 blank"> </span>of <span class="_35 blank"> </span>the <span class="_35 blank"> </span>resultant <span class="_35 blank"> </span>is <span class="_35 blank"> </span>negative <span class="_35 blank"> </span>and <span class="_35 blank"> </span>the <span class="_35 blank"> </span><span class="ff2 fs12 ls15">y</span> <span class="_35 blank"> </span>com<span class="_f blank"></span>ponent <span class="_35 blank"> </span>is <span class="_35 blank"> </span>positive, </span></div><div class="t m0 x1 h33 y88 ff1 fs15 fc0 sc0 lsb wsd">characteristic <span class="_36 blank"> </span>of <span class="_36 blank"> </span>the <span class="_36 blank"> </span>second <span class="_1b blank"> </span>qu<span class="_f blank"></span>adrant, <span class="_36 blank"> </span>we <span class="_36 blank"> </span>find <span class="_1b blank"> </span>the <span class="_36 blank"> </span>angle <span class="_36 blank"> </span>is <span class="_36 blank"> </span>132° <span class="_36 blank"> </span>(measured </div><div class="t m0 x1 h33 y89 ff1 fs15 fc0 sc0 lsb wsd">counterclockwise fr<span class="_f blank"></span>om +<span class="ff2 fs12 ls15 v0">x</span><span class="v0"> axis). </span></div><div class="t m0 x1 h33 y8a ff1 fs15 fc0 sc0 lsb wsd">9. <span class="_d blank"> </span>We <span class="_d blank"> </span>write </div><div class="c x4f y8b w8 h42"><div class="t m0 x6 h22 y8c ff3 fs10 fc0 sc0 lsb">G</div></div><div class="c x50 y8b w8 h42"><div class="t m0 x6 h22 y8c ff3 fs10 fc0 sc0 lsb">G</div></div><div class="c x51 y8d wf h43"><div class="t m0 x6 h22 y8e ff3 fs10 fc0 sc0 lsb">G</div></div><div class="c x4f y8f wb h2d"><div class="t m0 x6 h23 y4d ff2 fs10 fc0 sc0 lsb">r</div></div><div class="c x6 yd w4 hd"><div class="t m0 x52 h23 y90 ff2 fs10 fc0 sc0 lsb ws2a">a b<span class="_37 blank"></span><span class="ff9 ws2b">=<span class="_38 blank"> </span>+ <span class="ff1 fs15 wsd">. <span class="_d blank"> </span>When <span class="_d blank"> </span>not <span class="_d blank"> </span>explicitly <span class="_e blank"> </span>displayed, <span class="_d blank"> </span>the <span class="_d blank"> </span>units <span class="_d blank"> </span>here <span class="_e blank"> </span>are <span class="_d blank"> </span>assumed <span class="_d blank"> </span>to <span class="_d blank"> </span>be </span></span></div><div class="t m0 x1 h33 y91 ff1 fs15 fc0 sc0 lsb wsd">meters. </div><div class="t m0 x1 h33 y92 ff1 fs15 fc0 sc0 lsb wsd">(a) The <span class="ff2 fs12 ls15 v0">x</span><span class="v0"> and the <span class="ff2 fs12 ls16">y</span> components of </span></div></div><div class="c x26 y93 w5 h44"><div class="t m0 x6 h22 y94 ff3 fs10 fc0 sc0 lsb">G</div></div><div class="c x26 y95 w10 h45"><div class="t m0 x6 h23 y96 ff2 fs10 fc0 sc0 lsb">r</div></div><div class="t m0 x53 h33 y97 ff1 fs15 fc0 sc0 lsb wsd"> are <span class="ff2 fs12 ls19">r<span class="fs8 lsb wsb v5">x</span></span> = <span class="ff2 fs12 ws1c">a<span class="fs8 wsb v5">x</span></span> + <span class="ff2 fs12 ws1c">b<span class="fs8 wsb v5">x</span></span> = <span class="_1 blank"> </span>(4.0 m) ! (13 m) = !9.0 m and <span class="ff2 fs12 ls19">r<span class="fs8 lsb wsb v5">y</span></span> <span class="_1 blank"> </span>= </div><div class="t m0 x1 h46 y98 ff2 fs12 fc0 sc0 lsb ws1c">a<span class="fs8 wsb v5">y</span><span class="ff1 fs15 wsd"> + </span>b<span class="fs8 wsb v5">y</span><span class="ff1 fs15 wsd"> = (3.0 m<span class="_f blank"></span>) + (7.0 m) <span class="_f blank"></span>= 10 m, res<span class="_f blank"></span>pectively. Thus <span class="_39 blank"> </span><span class="ffa ws2c v8">\u2022 \u2022</span></span></div><div class="t m0 x1e h33 y98 ffa fs15 fc0 sc0 lsb ws2d">(<span class="_36 blank"> </span>9.0 m) i<span class="_38 blank"> </span>(<span class="_3a blank"></span>10 m)<span class="_35 blank"> </span>j<span class="_3b blank"></span><span class="ff2 fs12 ls24">r<span class="ff9 fs15 lsb ws2e">= \u2212<span class="_3c blank"> </span>+</span></span></div><div class="t m0 x54 h2a y99 ff3 fs12 fc0 sc0 ls25">G<span class="ffa fs15 lsb v1">.</span></div><div class="t m0 x1 h33 y9a ffa fs15 fc0 sc0 lsb wsd">(b) The mag<span class="_f blank"></span>nitude of </div><div class="c x55 y9b w5 h47"><div class="t m0 x6 h2a y9c ff3 fs12 fc0 sc0 lsb">G</div></div><div class="c x55 y9d w11 h3f"><div class="t m0 x6 h2e y9e ff2 fs12 fc0 sc0 lsb">r</div></div><div class="t m0 x13 h33 y9a ffa fs15 fc0 sc0 lsb ws2f">is</div><div class="t m0 x1b h48 y9f ffa fs19 fc0 sc0 lsb ws30">2 2<span class="_3d blank"> </span>2<span class="_3e blank"> </span>2</div><div class="t m0 x42 h33 ya0 ffa fs15 fc0 sc0 lsb wsd">|<span class="_3f blank"> </span>|<span class="_40 blank"> </span>(<span class="_36 blank"> </span>9.0<span class="_1 blank"> </span> m)<span class="_8 blank"> </span>(<span class="_41 blank"></span>10<span class="_1 blank"> </span> m)<span class="_13 blank"> </span>13 <span class="_f blank"></span>m</div><div class="t m0 x56 h1c ya1 ff2 fse fc0 sc0 lsb ws31">x y</div><div class="t m0 x57 h2e ya0 ff2 fs12 fc0 sc0 lsb ws32">r r<span class="_42 blank"> </span>r<span class="_43 blank"> </span>r<span class="_44 blank"></span><span class="ff9 fs15 ws33">= =<span class="_42 blank"> </span>+<span class="_30 blank"> </span>=<span class="_45 blank"> </span>\u2212<span class="_46 blank"> </span>+<span class="_47 blank"> </span>=</span></div><div class="t m0 x21 h2a ya2 ff3 fs12 fc0 sc0 ls26">G<span class="ffa fs15 lsb v1">.</span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><div class="t m0 x16 h33 ya3 ffa fs15 fc0 sc0 lsb wsd">11. <span class="_d blank"> </span>We <span class="_16 blank"> </span>find <span class="_d blank"> </span>the <span class="_d blank"> </span>components <span class="_16 blank"> </span>and <span class="_d blank"> </span>then <span class="_d blank"> </span>add <span class="_d blank"> </span>them <span class="_16 blank"> </span>(as <span class="_d blank"> </span>scalars, <span class="_d blank"> </span>not <span class="_d blank"> </span>vectors). <span class="_16 blank"> </span>With <span class="_d blank"> </span><span class="ff2 fs12 ws1c">d</span> <span class="_d blank"> </span>= <span class="_d blank"> </span>3.40 </div><div class="t m0 x16 h33 ya4 ffa fs15 fc0 sc0 lsb wsd">km and </div><div class="t m2 x58 h1e ya4 ff4 fsf fc0 sc0 lsb ws16">\u03b8</div><div class="t m0 x36 h33 ya4 ffa fs15 fc0 sc0 lsb wsd"> = 35.0° we fin<span class="_f blank"></span>d <span class="ff2 fs12 ws1c">d</span> cos </div><div class="t m2 x44 h1e ya4 ff4 fsf fc0 sc0 lsb ws16">\u03b8</div><div class="t m0 x59 h33 ya4 ffa fs15 fc0 sc0 lsb wsd"> + <span class="ff2 fs12 ws1c">d</span> sin </div><div class="t m2 x5a h1e ya4 ff4 fsf fc0 sc0 lsb ws16">\u03b8</div><div class="t m0 x32 h33 ya4 ffa fs15 fc0 sc0 lsb wsd"> = 4.74 km<span class="_f blank"></span>. </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><div class="t m0 x1 h33 ya5 ffa fs15 fc0 sc0 lsb wsd">13. All distances in this s<span class="_f blank"></span>olution are understood to <span class="_f blank"></span>be in meters. </div><div class="t m0 x1 h32 ya6 ffa fs15 fc0 sc0 lsb ws34">(a) <span class="ws35 v8">\u2022<span class="_48 blank"> </span>\u2022<span class="_49 blank"> </span>\u2022<span class="_4a blank"> </span>\u2022 \u2022<span class="_4b blank"> </span>\u2022</span></div><div class="t m0 x2a h33 ya6 ffa fs15 fc0 sc0 lsb wsd">[4.0<span class="_4c blank"> </span>(<span class="_36 blank"> </span>1.0)]<span class="_0 blank"> </span>i<span class="_4d blank"> </span>[(<span class="_36 blank"> </span>3.0)<span class="_4e blank"> </span>1.0]<span class="_0 blank"> </span>j<span class="_4d blank"> </span>(<span class="_41 blank"></span>1.0<span class="_4c blank"> </span>4.0)<span class="_16 blank"> </span>k<span class="_4d blank"> </span>(3.0<span class="_d blank"> </span>i<span class="_3f blank"> </span>2.0<span class="_d blank"> </span>j<span class="_22 blank"> </span>5.0<span class="_0 blank"> </span>k) m<span class="_f blank"></span>.<span class="_4f blank"></span><span class="ff2 fs12 ws36">a b<span class="_50 blank"></span><span class="ffb fs15 ws37">+<span class="_3f blank"> </span>=<span class="_51 blank"> </span>+ \u2212<span class="_52 blank"> </span>+<span class="_3f blank"> </span>\u2212<span class="_2 blank"> </span>+<span class="_4b blank"> </span>+<span class="_53 blank"> </span>+<span class="_52 blank"> </span>=<span class="_54 blank"> </span>\u2212<span class="_55 blank"> </span>+</span></span></div><div class="t m0 x5b h49 ya7 ff3 fsa fc0 sc0 lsb">G</div><div class="t m0 x5c h49 ya8 ff3 fsa fc0 sc0 lsb">G</div><div class="t m0 x1 h4a ya9 ffa fs15 fc0 sc0 lsb ws38">(b) <span class="ws39 v8">\u2022<span class="_56 blank"> </span>\u2022<span class="_31 blank"> </span>\u2022<span class="_57 blank"> </span>\u2022 \u2022<span class="_4b blank"> </span>\u2022</span></div><div class="t m0 x28 h33 ya9 ffa fs15 fc0 sc0 lsb wsd">[<span class="_1 blank"> </span>4.0<span class="_58 blank"> </span>(<span class="_36 blank"> </span>1.0)]<span class="_16 blank"> </span>i<span class="_4d blank"> </span>[(<span class="_36 blank"> </span>3.0)<span class="_10 blank"> </span>1.0]<span class="_d blank"> </span>j<span class="_3f blank"> </span>(<span class="_41 blank"></span>1.0<span class="_4c blank"> </span>4.0)<span class="_16 blank"> </span>k<span class="_4d blank"> </span>(5.0<span class="_35 blank"> </span>i<span class="_3f blank"> </span>4.0<span class="_0 blank"> </span>j<span class="_4e blank"> </span>3.0<span class="_35 blank"> </span>k)<span class="_f blank"></span> m<span class="_3a blank"></span>.<span class="_59 blank"></span><span class="ff2 fs12 ws3a">a b<span class="_50 blank"></span><span class="ffb fs15 ws3b">\u2212<span class="_3f blank"> </span>=<span class="_51 blank"> </span>\u2212 \u2212<span class="_5a blank"> </span>+<span class="_3f blank"> </span>\u2212<span class="_2 blank"> </span>\u2212<span class="_5b blank"> </span>+<span class="_53 blank"> </span>\u2212<span class="_52 blank"> </span>=<span class="_5c blank"> </span>\u2212<span class="_42 blank"> </span>\u2212</span></span></div><div class="t m0 x5b h49 yaa ff3 fsa fc0 sc0 lsb">G</div><div class="t m0 x5c h49 yab ff3 fsa fc0 sc0 lsb">G</div><div class="t m0 x1 h33 yac ffa fs15 fc0 sc0 lsb wsd">(c) <span class="_0 blank"> </span>The <span class="_35 blank"> </span>requir<span class="_f blank"></span>ement </div><div class="c x5d yad w5 h4b"><div class="t m0 x6 h49 yae ff3 fsa fc0 sc0 lsb">G</div></div><div class="c x46 yaf w8 h4c"><div class="t m0 x6 h49 yb0 ff3 fsa fc0 sc0 lsb">G</div></div><div class="c x23 yad w5 h4b"><div class="t m0 x6 h49 yae ff3 fsa fc0 sc0 lsb">G</div></div><div class="t m0 x5e h33 yb1 ff2 fs1a fc0 sc0 lsb ws3c">a b<span class="_38 blank"> </span>c<span class="_5d blank"></span><span class="ffb ws3d">\u2212<span class="_4d blank"> </span>+<span class="_1e blank"> </span>= <span class="ffa fs15 wsd">0<span class="_0 blank"> </span> leads <span class="_0 blank"> </span>to </span></span></div><div class="c x33 yb2 wc h4d"><div class="t m0 x6 h49 yb3 ff3 fsa fc0 sc0 lsb">G</div></div><div class="c xc yb4 w5 h4c"><div class="t m0 x6 h49 yb5 ff3 fsa fc0 sc0 lsb">G</div></div><div class="c x5f yb2 w5 h4d"><div class="t m0 x6 h49 yb3 ff3 fsa fc0 sc0 lsb">G</div></div><div class="c x6 yd w4 hd"><div class="t m0 x33 h33 yb6 ff2 fs1a fc0 sc0 lsb ws3e">c<span class="_21 blank"> </span>b a<span class="_37 blank"></span><span class="ffb ws3f">=<span class="_1e blank"> </span>\u2212 <span class="ffa fs15 wsd">,<span class="_0 blank"> </span> which <span class="_0 blank"> </span>we <span class="_0 blank"> </span>note <span class="_0 blank"> </span>is <span class="_35 blank"> </span>the<span class="_f blank"></span> <span class="_0 blank"> </span>opposite <span class="_0 blank"> </span>of </span></span></div><div class="t m0 x1 h4e yb7 ffa fs15 fc0 sc0 lsb wsd">what we found in <span class="_f blank"></span>part (b). Thus, <span class="_5e blank"> </span><span class="ws40 v8">\u2022 \u2022<span class="_5f blank"> </span>\u2022</span></div><div class="t m0 xd h33 yb7 ffa fs15 fc0 sc0 lsb wsd">(<span class="_36 blank"> </span>5.0<span class="_e blank"> </span>i <span class="_1e blank"> </span> 4.0<span class="_35 blank"> </span>j <span class="_1e blank"> </span> 3.0<span class="_0 blank"> </span>k) <span class="_f blank"></span>m.</div><div class="t m0 x15 h4f yb8 ff2 fs1a fc0 sc0 ls27">c<span class="ffb lsb ws41">= \u2212<span class="_60 blank"> </span>+<span class="_4a blank"> </span>+</span></div><div class="t m0 x59 h49 yb9 ff3 fsa fc0 sc0 lsb">G</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf9" class="pf w0 h0" data-page-no="9"><div class="pc pc9 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x60 yba w12 h50" alt="" src="https://files.passeidireto.com/c9399b22-093a-404c-9c9a-1c9407e09aa7/bg9.png"><div class="c x6 yd w4 hd"><div class="t m0 x1 h33 ybb ffa fs15 fc0 sc0 lsb wsd">(a) Along the <span class="ff2 fs1a ls28">x</span> axis, w<span class="_f blank"></span>e have (with the centim<span class="_f blank"></span>eter unit understood<span class="_f blank"></span>) </div><div class="t m0 x2c h33 ybc ffa fs15 fc0 sc0 lsb ws42">30.0<span class="_61 blank"> </span>20.0 80.0<span class="_51 blank"> </span>140,</div><div class="t m0 xd h1c ybd ff2 fse fc0 sc0 lsb">x</div><div class="t m0 x61 h4f ybe ff2 fs1a fc0 sc0 lsb ws43">b<span class="_62 blank"></span><span class="ffc ws44">+<span class="_45 blank"> </span>\u2212<span class="_4b blank"> </span>\u2212<span class="_54 blank"> </span>= \u2212</span></div><div class="t m0 x1 h51 ybf ffa fs15 fc0 sc0 lsb wsd">which gives <span class="ff2 fs1a ws43">b<span class="fs8 wsb v5">x</span></span><span class="ff1 v0"> = !<span class="_f blank"></span>70.0 cm. </span></div><div class="t m0 x1 h33 yc0 ff1 fs15 fc0 sc0 lsb wsd">(b) Along the <span class="ff2 fs1a ls28 v0">y</span><span class="v0"> ax<span class="_f blank"></span>is we have </span></div><div class="t m0 x62 h33 yc1 ff1 fs15 fc0 sc0 lsb ws45">4 0 . 0<span class="_45 blank"> </span>7 0 . 0<span class="_63 blank"> </span>7 0 . 0<span class="_54 blank"> </span>2 0 . 0</div><div class="t m0 x63 h52 yc2 ff2 fs1b fc0 sc0 lsb">y</div><div class="t m0 x64 h4f yc3 ff2 fs1a fc0 sc0 lsb ws43">c<span class="_64 blank"></span><span class="ffc ws46">\u2212<span class="_65 blank"> </span>+<span class="_66 blank"> </span>\u2212<span class="_67 blank"> </span>= \u2212</span></div><div class="t m0 x1 h53 yc4 ff1 fs15 fc0 sc0 lsb wsd">which yields <span class="ff2 fs1a ls29">c<span class="fs8 lsb wsb v5">y</span></span><span class="v0"> = 80<span class="_f blank"></span>.0 cm. </span></div><div class="t m0 x1 h54 yc5 ff1 fs15 fc0 sc0 lsb wsd">(c) The magnit<span class="_f blank"></span>ude of the final location (<span class="_f blank"></span>!140 , !20.0) is <span class="_47 blank"> </span><span class="fs1c ws47 v6">2 2</span></div><div class="t m0 x65 h33 yc6 ff1 fs15 fc0 sc0 lsb wsd">(<span class="_36 blank"> </span>140)<span class="_68 blank"> </span>(<span class="_36 blank"> </span>20.0)<span class="_43 blank"> </span>141<span class="_f blank"></span> cm<span class="_3a blank"></span>.</div><div class="t m0 x1f h55 yc7 ffc fs1a fc0 sc0 lsb ws48">\u2212<span class="_5a blank"> </span>+ \u2212<span class="_69 blank"> </span>=</div><div class="t m0 x1 h33 yc8 ff1 fs15 fc0 sc0 lsb wsd">(d) <span class="_16 blank"> </span>Since <span class="_16 blank"> </span>the <span class="_16 blank"> </span>displacement <span class="_16 blank"> </span>is <span class="_16 blank"> </span>in <span class="_16 blank"> </span>the <span class="_16 blank"> </span>third <span class="_16 blank"> </span>quadrant, <span class="_16 blank"> </span>the <span class="_16 blank"> </span>angle <span class="_d blank"> </span>of <span class="_1 blank"> </span>the <span class="_16 blank"> </span>overall <span class="_16 blank"> </span>displacement </div><div class="t m0 x1 h33 yc9 ff1 fs15 fc0 sc0 lsb wsd">is <span class="_16 blank"> </span>given <span class="_16 blank"> </span>by </div><div class="t m2 x11 h56 yca ffc fsf fc0 sc0 lsb">\u03c0</div><div class="t m0 x4f h57 yca ff1 fs15 fc0 sc0 lsb wsd"> <span class="_16 blank"> </span>+<span class="_6a blank"> </span><span class="fs19 v6">1</span></div><div class="t m0 x66 h33 yca ff1 fs15 fc0 sc0 lsb ws49">t a n<span class="_10 blank"> </span>[ (<span class="_6 blank"> </span>2 0 . 0)<span class="_0 blank"> </span>/ (<span class="_6 blank"> </span>1 4 0) ]</div><div class="t m0 x67 h58 ycb ffc fs3 fc0 sc0 ls2a">\u2212<span class="fs1a lsb ws4a v9">\u2212<span class="_6b blank"> </span>\u2212 <span class="ff1 fs15 wsd">or <span class="_16 blank"> </span>188° <span class="_16 blank"> </span>counterclockwise <span class="_16 blank"> </span>from <span class="_16 blank"> </span>the <span class="_16 blank"> </span>+<span class="ff2 fs1a ls29">x</span> <span class="_d blank"> </span>axi<span class="_f blank"></span>s <span class="_16 blank"> </span>(172° </span></span></div><div class="t m0 x1 h33 ycc ff1 fs15 fc0 sc0 lsb wsd">clockwise from<span class="_f blank"></span> the +<span class="ff2 fs1a ls2b v0">x</span><span class="v0"> axis).</span></div><div class="t m0 x1 h33 ycd ff1 fs15 fc0 sc0 lsb wsd">15. <span class="_35 blank"> </span>Reading <span class="_6c blank"> </span>carefully, <span class="_6c blank"> </span>we <span class="_6c blank"> </span>see <span class="_35 blank"> </span>that <span class="_35 blank"> </span>the <span class="_6c blank"> </span>(</div><div class="t m0 x64 h33 yce ff2 fs1a fc0 sc0 lsb wsd">x, <span class="_6c blank"> </span>y<span class="ff1 fs15">) <span class="_35 blank"> </span>specifications <span class="_6c blank"> </span>for <span class="_6c blank"> </span>each <span class="_35 blank"> </span>"dart# <span class="_6c blank"> </span>are <span class="_6c blank"> </span>to <span class="_35 blank"> </span>be </span></div><div class="t m0 x1 h33 ycf ff1 fs15 fc0 sc0 lsb wsd">interpreted <span class="_17 blank"> </span>as (<span class="_2b blank"> </span>,<span class="_32 blank"> </span>)<span class="_6d blank"></span><span class="ffc fs1a ws4b">\u2206 \u2206<span class="_6e blank"></span><span class="ff2 ws4c">x y<span class="_6f blank"></span><span class="ff1 fs15 wsd"> <span class="_70 blank"> </span> descriptions of <span class="_17 blank"> </span>the <span class="_26 blank"> </span>corr<span class="_f blank"></span>esponding displacement <span class="_17 blank"> </span>vectors. <span class="_17 blank"> </span>We </span></span></span></div><div class="t m0 x1 h33 yd0 ff1 fs15 fc0 sc0 lsb wsd">combine the d<span class="_f blank"></span>ifferent parts of this pro<span class="_f blank"></span>blem into a <span class="_f blank"></span>single exposition. </div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pfa" class="pf w0 h0" data-page-no="a"><div class="pc pca w0 h0"><img fetchpriority="low" loading="lazy" class="bi x48 yd1 w13 h50" alt="" src="https://files.passeidireto.com/c9399b22-093a-404c-9c9a-1c9407e09aa7/bga.png"><div class="t m0 x1 h59 yd2 ffa fs1d fc0 sc0 lsb wsd">21. <span class="_1 blank"> </span>(a) <span class="_16 blank"> </span>With i</div><div class="t m0 x68 h5a yd3 ffa fs8 fc0 sc0 lsb wsb">^<span class="fs1d wsd v9"> directed <span class="_1 blank"> </span>forward <span class="_1 blank"> </span>an<span class="_1 blank"> </span>d j</span></div><div class="t m0 x61 h5a yd3 ffa fs8 fc0 sc0 lsb wsb">^<span class="fs1d wsd v9"> directed <span class="_1 blank"> </span>leftward, <span class="_16 blank"> </span>then <span class="_1 blank"> </span>the <span class="_16 blank"> </span>resultant <span class="_1 blank"> </span>is <span class="_16 blank"> </span>(5.00 i</span></div><div class="t m0 x69 h5a yd3 ffa fs8 fc0 sc0 lsb wsb">^<span class="fs1d wsd v9"> + <span class="_1 blank"> </span>2.00 </span></div><div class="t m0 x1 h59 yd4 ffa fs1d fc0 sc0 lsb">j</div><div class="t m0 x1 h5a yd5 ffa fs8 fc0 sc0 ls2c">^<span class="fs1d lsb wsd v9">) <span class="_0 blank"> </span>m <span class="_0 blank"> </span>. <span class="_0 blank"> </span>The <span class="_0 blank"> </span>magnitude <span class="_0 blank"> </span>is <span class="_0 blank"> </span>given <span class="_0 blank"> </span>by <span class="_0 blank"> </span>the <span class="_0 blank"> </span>Pythagorea<span class="_1 blank"> </span>n <span class="_0 blank"> </span>theorem: <span class="_1a blank"> </span></span><span class="fs19 lsb ws4d v0">2 2</span></div><div class="t m0 x6a h59 yd6 ffa fs1d fc0 sc0 lsb wsd">(5.00<span class="_1 blank"> </span> m<span class="_1 blank"> </span>)<span class="_45 blank"> </span>(2.00<span class="_1 blank"> </span> m<span class="_1 blank"> </span>)<span class="_71 blank"></span><span class="ff7 ls2d">+<span class="ffa lsb"> = </span></span></div><div class="t m0 x1 h59 yd7 ffa fs1d fc0 sc0 lsb wsd">5.385 m <span class="ffd ls2e">\u2248</span> 5.39 m. </div><div class="t m0 x1 h5b yd8 ffa fs1d fc0 sc0 lsb wsd">(b) The angle is tan<span class="ffd fs8 wsb v6">\u2212<span class="ffa ls2f">1</span></span><span class="ls30 v0">(<span class="fs1e lsb ws4e">2.00/<span class="_1 blank"> </span>5.00</span><span class="ls31">)<span class="ffd ls32">\u2248</span><span class="lsb"> 21.8º (left of forward).</span></span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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