12.2 Videos Guide

12.2a

• Vector from $P\left(x_{1},y_{1},z_{1}\right)$ to $Q(x_{2},y_{2},z_{2})$
  (coordinates of terminal point) $-$ (coordinates of initial point)

  • $\mathbf{a}=\left\langle x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1}\right\rangle$

12.2b

• Vector arithmetic: let $\mathbf{a}=\left\langle a_{1},a_{2},a_{3}\right\rangle$, $\mathbf{b}=\left\langle b_{1},b_{2},b_{3}\right\rangle$, and $c$ be a real number

  • Scalar multiplication: $c\mathbf{a}=\left\langle {ca}_{1},{ca}_{2},{ca}_{3}\right\rangle$

12.2c

• Addition/subtraction: $\mathbf{a}\pm\mathbf{b}=\langle a_{1}\pm b_{1},a_{2}\pm b_{2},a_{3}\pm b_{3}\rangle$

12.2d

• Magnitude of a vector $\mathbf{a}=\left\langle a,\ b,\ c\right\rangle$

  • $\left|\mathbf{a}\right|=\sqrt{a^{2}+b^{2}+c^{2}}$

• Standard basis vectors for $V_{3}$

  • $\mathbf{i}=\left\langle 1,\ 0,\ 0\right\rangle$

  • $\mathbf{j}=\left\langle 0,\ 1,\ 0\right\rangle$

  • $\mathbf{k}=\left\langle 0,\ 0,\ 1\right\rangle$

Exercise:

• Find a unit vector that has the same direction as the vector $-5\mathbf{i}+3\mathbf{j}-\mathbf{k}$.