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

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Answer 1 · 2016-07-26T05:38:10

#75.9^o# or

#360^o-75.9^o=284.1^o#.

Any two vectors #p and q# determine a plane. For the purpose of

the visualizing the angle in-between #p 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 (a)=+-|p X q |/( | p | | q | }#

Here, #p = (6, 2, -4) and q = (2, 8, 3)#

So, # |p X q |=|( 38, -2 6, 44)|= sqrt 4056,#

# | p | =sqrt 56 and | q | = sqrt 77#.

And so, #sin a =+-sqrt (2056/((56)(77)))=+-sqrt (507/539)#.

Thus, #a = arc sin sqrt ( 507/539 )=#75.9^o# or

#360^o-75.9^o=284.1^o#. for the negative sign.