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

Emeli Hagan

Emeli Hagan

Answered question

2021-05-28

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

Answer & Explanation

Layton

Layton

Skilled2021-05-29Added 89 answers

Write the vector as ordered triples
i+j=<1,1,0>
i+k=<1,0,1>
Let  be the vector orthogonal to both
The dot product of two vectors equals 0 if they are orthogonal.
<1,1,0>=(1)x+(1)y+(0)z=x+y=0
<1,1,0>=(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.
<1,1,1>
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
|<1,1,1>|=12+(1)2+(1)2=3
then divide each component by the length, which turns it into a unit vector
<13,13,13>
This kind of problem is a bit easier using the cross product that comes in the next section.
Result
<13,13,13> or <13,13,13>

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