Para se calcular o volume de um tetraedro, basta-nos calcular o produto misto dos vetores dados e multiplicar por 1/6:

\(\begin{align} V&={1\over6}\left\vert(\vec{OA}\times\vec{OB})\cdot\vec{OC}\right\vert\\ &={1\over6}\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\1&y&2\\2&0&1\end{vmatrix}\cdot(0,3,1)\\ &={1\over6}(y,3,-2y)\cdot(0,3,1)\\ &={1\over6}(9-2y)\\ \end{align}\)

Além disso, o volume também pode ser calculado por:

\(\begin{align} V&={1\over3}A_bh\\ &={1\over3}\left\vert\vec{OB}\times\vec{OC}\right\vert h\\ &={1\over3}\left\vert\left\vert\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&0&1\\0&3&1\end{vmatrix}\right\vert\right\vert\cdot{1\over7}\\ &={1\over21}\left\vert\left\vert(-3,-2,6)\right\vert\right\vert\\ &={1\over21}\sqrt{3^2+2^2+6^2}\\ &={1\over21}\cdot7\\ &={1\over3}\\ \end{align}\)

Igualando as duas expressões pro volume, temos:

\({1\over6}(9-2y)={1\over3}\Rightarrow \boxed{y={7\over2}}\)