How do you find the unit vector which bisects the angle AOB given A and B are position vectors a=2i-2j-k and b=3i+4k?

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Answer 1 · 2016-12-08T22:56:43

Please see the explanation.

Given: #bara = 2hati - 2hatj - hatk and barb = 3hati + 4hatk#

Here is a reference regarding how to find an Angle Bisector Vector , #barc#:

#barc = ||bara||barb + ||barb||bara#

Compute the magnitude of #bara#:

#||bara|| = sqrt(2^2 + (-2)^2 + (-1)^2) = sqrt(9) = 3#

Compute the magnitude of #barb#:

#||barb|| = sqrt(3^2 + 4^2) = sqrt(25) = 5#

#||bara||barb = 3(3hati + 4hatk) = 9hati + 12hatk#

#||barb||bara = 5(2hati - 2hatj - hatk) = 10hati - 10hatj - 5hatk#

#barc = 9hati + 12hatk + 10hati - 10hatj - 5hatk#

#barc = 19hati - 10hatj + 7hatk#

However, this is not the unit vector, #hatc#. To make is at unit vector we must divide vector #barc# by its magnitude:

||barc|| = sqrt(19^2 + (-10)^2 + 7^2) = sqrt(510)

#hatc = 19/sqrt(510)hati - 10/sqrt(510)hatj + 7/sqrt(510)hatk#