Find a unit vector that is orthogonal to both i+j and i+k.

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Layton · Accepted Answer

Write the vector as ordered triples i+j=&amp;lt;1,1,0&amp;gt; i+k=&amp;lt;1,0,1&amp;gt; Let  be the vector orthogonal to both The dot product of two vectors equals 0 if they are orthogonal. &amp;lt;1,1,0&amp;gt;⋅=(1)x+(1)y+(0)z=x+y=0 &amp;lt;1,1,0&amp;gt;⋅=(1)x+(1)y+(0)z=x+z=0 Solve the system. There are 3 unknowns but only two equations, which just means the solution is not unique. x+y=0 x+z=0 x+y=x+z y=z y=−x z=−x What we can do here is pick a number for x, like x=1, then y=-1 and z=-1. &amp;lt;1,−1,−1&amp;gt; which gives us one possible vector orthogonal to both. They ask for a unit vector so we have to normalize it. First find the length of our vector so for |&amp;lt;1,−1,−1&amp;gt;|=12+(−1)2+(−1)2=3 then divide each component by the length, which turns it into a unit vector &amp;lt;−13,13,13&amp;gt; This kind of problem is a bit easier using the cross product that comes in the next section. Result &amp;lt;13,−13,−13&amp;gt; or &amp;lt;−13,13,13&amp;gt;

Find a unit vector that is orthogonal to both i+j and i+k.

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