Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 1

In components, the scalar triple product isIt is the determinant with rows $\mathbf{a},\mathbf{b},\mathbf{c}$. A cyclic permutation of the rows has positive sign, so $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\mathbf{c}\cdot (\mathbf{a}\times \mathbf{b})=(\mathbf{a}\times \mathbf{b})\cdot \mathbf{c}$. Equivalently the Levi-Civita symbol expression ${ϵ}_{ijk}{a}_i{b}_j{c}_k$ is unchanged by that cyclic relabeling.

Take $P=(2,-1,2)$, $Q=(1,2,3)$ and $R=(4,1,0)$. The two plane directions are $Q-P=(-1,3,1)$ and $R-P=(2,2,-2)$. Their cross product is $(-8,0,-8)$, so a normal vector is $(1,0,1)$. The Cartesian equation of a plane is thereforeEach given point satisfies it, and the nonzero cross product confirms that the points determine a unique plane.

The volume of a parallelepiped is the absolute scalar triple product of its edge vectors. HereThe absolute value removes the orientation of the chosen edge ordering.