Chapter 9.2, Problem 19E

To determine

The perpendicularity of the given vector.

Answer to Problem 19E

The given vectors are perpendicular.

Explanation of Solution

Given :

  $\begin{array}{l}\text{u = }2\text{i - 8j}\\ \text{v = -12i}-3\text{j}\end{array}$

Calculation:

Definition of perpendicularity: Two vectors are said to be perpendicular if there dot product is zero;

i.e., $\text{u}\text{.v}=0$ .

Now,

  $\begin{array}{l}\text{u =}\langle {\text{a}}_{\text{1}}{\text{,b}}_{\text{1}}\rangle \text{and}\text{v =}\langle {\text{a}}_{\text{2}}{\text{,b}}_{\text{2}}\rangle \\ \text{u}{\text{.v = a}}_1{\text{a}}_2+{\text{b}}_{\text{1}}{\text{b}}_{\text{2}}\end{array}$

So,

  $\begin{array}{l}\text{u}\text{.v = }\left(2\right)\left(-12\right)+\left(-8\right)\left(-3\right)\\ \text{u}\text{.v = }-24+24\\ \text{u}\text{.v =}0\end{array}$

Here,

  $\text{u}\text{.v}=0$

Therefore, the given vectors are perpendicular.