12.4 Videos Guide

12.4a

• The cross product: Let $\mathbf{a}=\left\langle a_{1},a_{2},a_{3}\right\rangle$ and $\mathbf{b=}\left\langle b_{1},b_{2},b_{3}\right\rangle$

  • $\mathbf{a}\times\mathbf{b}=\left|\begin{matrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3} \end{matrix}\right|=\left(a_{2}b_{3}-a_{3}b_{2}\right)\mathbf{i}-\left(a_{1}b_{3}-a_{3}b_{1}\right)\mathbf{j}+\left(a_{1}b_{2}-a_{2}b_{1}\right)\mathbf{k}$

  • $\mathbf{a}\times\mathbf{b}$ is orthogonal to both $\mathbf{a}$ and $\mathbf{b}$

Exercise:

• Find the cross product $\mathbf{a}\times\mathbf{b}$ for $\mathbf{a}=\left\langle 2,\ -1,\ 5\right\rangle$ and $\mathbf{b}=\left\langle -4,\ 3,\ 8\right\rangle$.

12.4b

• Geometric applications and interpretations of the cross product

  • $\left|\mathbf{a}\times\mathbf{b}\right|=\left|\mathbf{a}\right|\left|\mathbf{b}\right|\sin\theta$, where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$

  • If $\mathbf{a}$ and $\mathbf{b}$ are parallel, then $\mathbf{a}\times\mathbf{b}=\mathbf{0}$, the zero vector

  • $\left|\mathbf{a}\times\mathbf{b}\right|$ gives the area of a parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$

  • Scalar triple product: $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$

    • $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|$ gives the volume of a parallelepiped with edges $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$

Exercises:

12.4c

• Find the area of the parallelogram with vertices $P(1,\ 0,\ 2)$, $Q(3,\ 3,\ 3)$, $R(7,\ 5,\ 8)$, and $S(5,\ 2,\ 7)$.

12.4d

• Find a nonzero vector orthogonal to the plane through the points $P(0,\ 0,\ -3)$, $Q(4,\ 2,\ 0)$, and $R(3,\ 3,\ 1)$, and (b) find the area of triangle $PQR$.