# Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. p(3,\ 0,\ 1) Q(-1,\ 2,\ 5) R(5,\ 1,\ -1) S(0,\ 4,\ 2)

Trigonometric equation and identitie
Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.
$$p(3,\ 0,\ 1)$$
$$Q(-1,\ 2,\ 5)$$
$$R(5,\ 1,\ -1)$$
$$S(0,\ 4,\ 2)$$

2021-05-13
Step 1
$$p(3,\ 0,\ 1),$$
$$Q(-1,\ 2,\ 5),$$
$$R(5,\ 1,\ -1),$$
$$S(0,\ 4,\ 2)$$
$$PQ=Q-P=\langle-4,\ 2,\ 4\rangle$$
$$PR=R-P=\langle2,\ 1,\ -2\rangle$$
$$PS=S-P =\langle-3,\ 4,\ 1\rangle$$
$$[PQ\ PR\ PS]=\left|\begin{matrix}-4 & 2 & 4 \\ 2 & 1 & -2 \\ -3 & 4 & 1\end{matrix}\right|$$
$$=-4(9)-2(-4)+4(11)$$
$$=-36+8+44=16$$
volume of the parallelepiped with adjacent edges PQ, PR, and $$PS=16$$ cubic units.