Question

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

Answer 1

The angle between two vectors depends on the sense of measurement. If `a^o` is the angle, for anticlockwise sense, `(360-a)^o` is the angle, for the clockwise sense.

The formula for the angle between any two vectors `P and Q` is

`arc sin( +-( | P X Q | ) / ( | P | |Q | ) )` .

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

`arc sin (+-sqrt (861 / ((170)(89))))`

`=arc sin (+-0.05691)`, nearly

`=3.2623^o`, nearly, for + sign.

For measurement in the opposite sense, the angle is

`=(360-3.2623)=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.