Subjective Type

How do you find the volume of the parallelepiped with adjacent edges $$pq,pr$$ and $$ps$$ where $$p(3,0,1),q(-1,2,5),r(5,1,-1)$$ and $$s(0,4,2)$$?

Solution

Given three vectors there is a product, called scalar triple product that gives the volume of parallelepiped that has three vectors as dimensions
$$\vec { PQ } =(3+1,0-2,1-5)=(4,-2,-4)$$
$$\vec { PR } =(3-5,0-1,1+1)=(-2,-1,2)$$
$$\vec { PS } =(3-0,0-4,1-2)=(3,-4,-1)$$
The scalar triple product is given by the determinant of the matrix $$(3\times 3)$$ that has in the rows the three components of the three vectors:
$$\left| +4-2-4 \right| $$
$$\left| -2-1+2 \right| $$
$$\left| +3-4-1 \right| $$
and the determinant is given with Laplace rule
$$4\cdot [(-1)(-1)-(2)(4)]-(-2)[(-2)(-1)-(2)\cdot (3)+(-4)[(-2)(-4)-(-1)(3)]=-16$$
So the volume $$V=\left| -16 \right| =16$$

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