# If A = <3 ,3 ,-7 >, B = <5 ,8 ,-4 > and C=A-B, what is the angle between A and C?

Oct 15, 2016

This is a special case of the dot-product; we know, without any further computation, that the angle $\theta$ between the two vectors is $\frac{\pi}{2}$

#### Explanation:

Given:
$\overline{A} = < 3 , 3 , - 7 >$,
$\overline{B} = < 5 , 8 , - 4 >$,
$\overline{C} = \overline{A} - \overline{B}$

Subtract the components of $\overline{B}$ from $\overline{A}$:

$\overline{C} = < \left(3 - 5\right) , \left(3 - 8\right) , \left(- 7 - - 4\right) >$

$\overline{C} = < - 2 , - 5 , - 3 >$

Compute $\overline{A} \cdot \overline{C}$ by multiplying the respective components:

$\overline{A} \cdot \overline{C} = \left(3\right) \left(- 2\right) + \left(3\right) \left(- 5\right) + \left(- 7\right) \left(- 3\right)$

$\overline{A} \cdot \overline{C} = 0$

This is a special case of the dot-product; we know, without any further computation, that the angle $\theta$ between the two vectors is $\frac{\pi}{2}$