Question

How do you find two unit vectors orthogonal to A=(1, 3, 0) B =(2, 0, 5)?

Answer 1

Please read the explanation.

Explanation:

Begin by computing the cross product. I use a determinant:

#barA xx barB = | (hati, hatj, hatk), (1, 3, 0), (2,0, 5) |#

`barA xx barB = hati|(3, 0),(0,5)| + hatj|(0,1),(5,2)| + hatk|(1,3),(2,0)|`

`barA xx barB = 15hati - 5hatj -6hatk`

Let `barC = 15hati - 5hatj -6hatk`

The unit vector, `hatC = barC/|barC|`

`|barC| = sqrt(15^2 + (-5^2) + (-6)^2)`

`|barC| = sqrt(286)`

`hatC = 15/sqrt(286)hati - 5/sqrt(286)hatj -6/sqrt(286)hatk`

The only other vector that can be orthogonal to `barA and barB` is:

`barB xx barA`

Because `barA xx barB = -(barB xx barA)`, the only other unit vector orthogonal to `barA and barB` is:

`-hatC = -15/sqrt(286)hati + 5/sqrt(286)hatj +6/sqrt(286)hatk`