How to find a unit vectors are orthogonal to both i+j and i+k?

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intacte87 · Accepted Answer

a=i+j=&amp;lt;1,1,0&amp;gt; and b=i+k=&amp;lt;1,0,1&amp;gt;    Then, let c=+&amp;lt;(cos⁡α,cos⁡β,cos⁡γ)&amp;gt; be the unit vectors orthogonal to a and b.    Thus, c.a=cos⁡α+cos⁡β=0    And c.b=cos⁡α+cos⁡γ=0    It follows that c=±&amp;lt;−cos⁡α,cos⁡α,cos⁡α&amp;gt;    And cos⁡α=±13    So, we have the answer: ±(−13,13,13)

Maria Lopez · Accepted Answer

Firstly find a vector V that is orthogonal to both, then make it a unit vector by taking

\frac{V}{| V |}

.
Finding the orthogonal vector is simple: just use the cross-product, (i+j) x (i+k).
Once you calculate that, just take its length and divide by that, and you'll naturally have a unit vector.

nick1337 · Accepted Answer

Just find a vector that is orthogonal to the vectors (1,1,0) and (1,0,1).  The dot product of two orthogonal vectors is 0. Therfore, if we call our new vector (x,y,z), we have: (x,y,z) dot (1,1,0) = x + y = 0 (x,y,z) dot (1,0,1) = x + z = 0 x = -y x= -z

How to find a unit vectors are orthogonal to both

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