Question

If Vector a= 2i - 4j + k and Vector b= -4i + j +2k how do you find a unit vector perpendicular to both vector a and b?

Answer 1

Please see the explanation.

Explanation:

Given:

`bara = 2hati - 4hatj + hatk and barb = -4hati + hatj + 2hatk`

A vector perpendicular to any two vectors is found by computing the cross-product:

#bara xx barb = | (hati, hatj, hatk, hati, hatj), (2,-4,1,2,-4), (-4, 1, 2, -4, 1) | = #

`hati{(-4)(2) - (1)(1)} + hatj{(1)(-4) - (2)(2)} + hatk{(2)(1)-(-4)(-4)} =`
`-9hati - 8hatj - 14hatk`
We can multiply by the scalar -1 and still have a perpendicular vector:

`9hati + 8hatj + 14hatk`

To obtain a unit vector, divide by the magnitude:

`|9hati + 8hatj + 14hatk| = sqrt(9^2 + 8^2 + 14^2) = sqrt(345)`

A unit vector perpendicular to both vectors is:

`9sqrt(345)/345hati + 8sqrt(345)/345hatj + 14sqrt(345)/345hatk`