12.3 Videos Guide

12.3a

• Definitions of the dot product: Let $\mathbf{a}=\left\langle a_{1},a_{2},a_{3}\right\rangle$ and $\mathbf{b=}\left\langle b_{1},b_{2},b_{3}\right\rangle$

  • $\mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$

    • Exercise:
      Find the dot product $\mathbf{a}\cdot\mathbf{b}$ of $\mathbf{a}=\left\langle 2,\ -4,\ 7\right\rangle$ and $\mathbf{b}=\left\langle -\frac{1}{2},\ 5,\ 13\right\rangle$.

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

    • Exercise:
      Find the angle between $\mathbf{a}$ and $\mathbf{b}$ from the previous exercise.

• Orthogonal vectors

If the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\theta=90{^{\circ}}$, then $\mathbf{a}\cdot\mathbf{b}=0$

12.3b

• Direction cosines: let $\alpha$, $\beta$, and $\gamma$ be angles made between a vector $\mathbf{a}=\left\langle a_{1},a_{2},a_{3}\right\rangle$ and the $x$-, $y$-, and $z$- axes, respectively

  • $\cos\alpha=\frac{a_{1}}{\left|\mathbf{a}\right|}$

  • $\cos\beta=\frac{a_{2}}{\left|\mathbf{a}\right|}$

  • $\cos\gamma=\frac{a_{3}}{\left|\mathbf{a}\right|}$

12.3c

• Scalar projection of $\mathbf{b}$ onto $\mathbf{a}$

  • ${\mbox{comp}}_{\mathbf{a}}\mathbf{b}=\frac{\mathbf{a}\cdot\mathbf{b}}{\left|\mathbf{a}\right|}$

• Vector projection of $\mathbf{b}$ onto $\mathbf{a}$

  • ${\mbox{proj}}_{\mathbf{a}}\mathbf{b}=\frac{\mathbf{a}\cdot\mathbf{b}}{\left|\mathbf{a}\right|^{2}}\mathbf{a}$

12.3d

Exercise:

• Find the scalar and vector projections of $\mathbf{b}$ onto $\mathbf{a}$.
  $\mathbf{a}=\left\langle -1,\ 4,\ 8\right\rangle$, $\mathbf{b}=\left\langle 12,\ 1,\ 2\right\rangle$