# What is the angle between < 6 , 2 , -4 >  and  < 2 , 8 , 3 > ?

Jul 26, 2016

${75.9}^{o}$ or

${360}^{o} - {75.9}^{o} = {284.1}^{o}$.

#### Explanation:

Any two vectors $p \mathmr{and} q$ determine a plane. For the purpose of

the visualizing the angle in-between $p \mathmr{and} q$, move the plane

parallel to itself to contain one of the vectors, and the other vector

can be moved parallel to itself such that the two are now co-

terminal.

If the angle between the two is ${a}^{o}$ for one sense, it is (360-

a)^o, for the opposite sense. So, the sign of sin a depends on the

sense of measurement. Importantly, anticlockwise sense with

respect to the North Pole is clockwise, with respect to the South

Pole.

Now, the formula for a is

$\sin \left(a\right) = \pm | p X q \frac{|}{| p | | q |}$

Here, $p = \left(6 , 2 , - 4\right) \mathmr{and} q = \left(2 , 8 , 3\right)$

So, $| p X q | = | \left(38 , - 2 6 , 44\right) | = \sqrt{4056} ,$

$| p | = \sqrt{56} \mathmr{and} | q | = \sqrt{77}$.

And so, $\sin a = \pm \sqrt{\frac{2056}{\left(56\right) \left(77\right)}} = \pm \sqrt{\frac{507}{539}}$.

Thus, $a = a r c \sin \sqrt{\frac{507}{539}} =$75.9^o# or

${360}^{o} - {75.9}^{o} = {284.1}^{o}$. for the negative sign.