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Find a unit vector u that is orthogonal to a and b where
a = −3 i − 8 j + 3 k and b = −7 i − 2 j + 5 k

u= ?

1 Answer
Jake M.
May 25, 2018

#u = 23/sqrt(1163) i + 3/sqrt(1163) j + 25/sqrt(1163) k #

Explanation:

The easiest way to find a 3D vector orthogonal to two others is a cross product. We can express this with a matrix
#[[i, j, k],[-3,-8,3],[-7,-2,5]] = i [[-8,3],[-2,5]] - j [[-3, 3],[-7, 5]] + k[[-3,-8],[-7,-2]]#
#= (-40 + 6) i + j(15 - 21) + k (6 - 56) #
#= -46i - 6 j - 50 k #
Since we want it to have unit magnitude, we can divide it by -2 just to clean up the numbers:
#23i + 3j + 25k #
To make unit magnitude, we only need to divide by the magnitude, which is
#||(23, 3, 25)|| = sqrt(529 + 9 + 625) = sqrt(1163) #

so the normalized vector is
#23/sqrt(1163) i + 3/sqrt(1163) j + 25/sqrt(1163) k #

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