Triple products and volumes practice problems
The scalar triple product u·(v × w): volumes of parallelepipeds, and testing for coplanar vectors. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 4, -4, -3 \right\rangle \), \( \displaystyle \left\langle -4, -1, -4 \right\rangle \) and \( \displaystyle \left\langle -16, 6, -2 \right\rangle \).
Are \( \displaystyle \mathbf u = \left\langle -1, 2, -1 \right\rangle \), \( \displaystyle \mathbf v = \left\langle 2, 3, 4 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle -2, -10, -6 \right\rangle \) coplanar?
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 4, 1, -4 \right\rangle \), \( \displaystyle \left\langle -4, 2, -2 \right\rangle \) and \( \displaystyle \left\langle -4, -3, -3 \right\rangle \).
Are \( \displaystyle \mathbf u = \left\langle 3, 4, 2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle 3, -2, 3 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle -1, 0, 0 \right\rangle \) coplanar?
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 0, -3, 3 \right\rangle \), \( \displaystyle \left\langle 1, 3, 3 \right\rangle \) and \( \displaystyle \left\langle 3, -1, 0 \right\rangle \).
Are \( \displaystyle \mathbf u = \left\langle 3, 0, 0 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -1, -4, 1 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle 2, 1, -3 \right\rangle \) coplanar?
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle -1, 4, 4 \right\rangle \), \( \displaystyle \left\langle 4, 3, -4 \right\rangle \) and \( \displaystyle \left\langle 1, -1, -1 \right\rangle \).
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 2, 2, 4 \right\rangle \), \( \displaystyle \left\langle 1, -2, -1 \right\rangle \) and \( \displaystyle \left\langle -4, 4, -1 \right\rangle \).
Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle -3, 1, -4 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle -1, -4, -3 \right\rangle \).
Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle -3, -1, -1 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle 0, -2, 4 \right\rangle \).