# 3.2 Post-audio exercises

## Example 42

Let **u** and **v** be the position vectors (6, 8) and (−12, 5), respectively.

**(a)**Sketch**u**and**v**on a single diagram. On the same diagram, sketch the projection of**u**onto**v**, and the projection of**v**onto**u**.**(b)**Determine the angle between**u**and**v**.**(c)**Determine the projection of**u**onto**v**, and the projection of**v**onto**u**.

### Answer

**(a)****(b)**Let*θ*denote the angle between**u**and**v**. Now**(c)**The projection of**u**onto**v**isThe projection of

**v**onto**u**is

## Example 43

Determine the angle between the vectors **u** = (3, 4, 5) and **v** = (1, 0, −1) in ^{3}.

### Answer

Let *θ* denote the angle between **u** and **v**. Now

and

Thus

## Example 44

Find a vector of length 2 that is perpendicular to both of the vectors **a** = (2, 1, 0) and **b** = (1, 0, −1).

### Answer

Let the vector we want be denoted by

**v** = (*x*,*y*,*z*),

for some real numbers *x*, *y* and *z*.

Since ‖**v**‖ = 2,

Since **v** is perpendicular to **a**,

thus

2*x* + *y* = 0,

that is,

Since **v** is perpendicular to **b**,

thus

*x* − *z* = 0,

that is,

Substituting the expressions for *y* in equation (S.4) and for *z* in equation (S.5) into equation (S.3), we obtain

*x*^{2} + (−2*x*)^{2} + *x*^{2} = 4;

thus

6*x*^{2} = 4,

so

It follows that the two possible vectors that satisfy the given conditions are