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

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Answer 1 · 2016-11-15T20:13:07

Please read the 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:

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