How do you find a unit vector u that is orthogonal to a and b where a = 7 i - 5 j + k and b = -7 j - 5 k?

1 Answer
Douglas K.
Nov 9, 2016

Please see the explanation for steps leading to the unit vector:

#hatc = 1/(5sqrt(158))(-18hati + 35hatj - 49hatk)#

Explanation:

The cross product of the two given vectors will give you a vector that is orthogonal; to make it a unit vector, you merely divide it by its magnitude.

#barc = bara xx barb = | (hati, hatj, hatk, hati, hatj), (7,5,1,7,5), (0,-7,-5,0,-7) | =#

#barc = hati{(5)(-5) - (1)(-7)} + hatj{(1)(0) - (7)(-5)} + hatk{(7)(-7) - (5)(0)} = #

#barc = -18hati + 35hatj - 49hatk#

The above vector, #barc# is orthogonal to both #bara and barb#.

To make #barc# a unit vector, compute its magnitude and then divide the vector by it:

#|barc| = sqrt((-18)^2 + 35^2 + (-49)^2)#

#|barc| = sqrt(324 +1225 + 2401)#

#|barc| = sqrt(3950) = 5sqrt(158)#

The unit vector is:

#hatc = 1/(5sqrt(158))(-18hati + 35hatj - 49hatk)#

Related questions
See all questions in Unit Vectors
Impact of this question
7676 views around the world
You can reuse this answer
Creative Commons License