# What is the angle between <8,5,9 > and < 4,3,8 >?

Jul 28, 2016

The angle between two vectors depends on the sense of measurement. If ${a}^{o}$ is the angle, for anticlockwise sense, ${\left(360 - a\right)}^{o}$ is the angle, for the clockwise sense.

The formula for the angle between any two vectors $P \mathmr{and} Q$ is

$a r c \sin \left(\pm \frac{| P X Q |}{| P | | Q |}\right)$ .

Here, # P = < 8, 5, 9 >, Q = < 4, 3, 8 >, P X Q = < 13, -26, 4 >, } P X Q } = sqrt 861, | P | = sqrt 170 and | Q | = sqrt 89.

The angle is

$a r c \sin \left(\pm \sqrt{\frac{861}{\left(170\right) \left(89\right)}}\right)$

$= a r c \sin \left(\pm 0.05691\right)$, nearly

$= {3.2623}^{o}$, nearly, for + sign.

For measurement in the opposite sense, the angle is

$= \left(360 - 3.2623\right) = {356.74}^{o}$, nearly, for the negative sign.

Note that anticlockwise rotation for northern latitudes, with respect

to North Pole, is clockwise rotation for the southern latitodes with

respect to the South Pole.