Question

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

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

Answers (1)

2021-05-13

Step 1
Let \(a=i+j,\ b=i+k\)
The cross product \(a\times b\) is arthogonal to both a and b
\(a\times b=\left|\begin{matrix}i & j & k \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}\right|\)
\(=i(1-0)-j(1-0)+k(0-1)\)
\(a\times b=i-j-k\)
Since the vector we found had magnitude \(\sqrt{3}\)
\(i.e.\ |a\times b|=\sqrt{\check{1}+\check{1}+\check{1}}=\sqrt{3}\)
A unit vector that is arthogonal to both
\(i+j\) and \(i+k\) is
\(u=\frac{(i-j-k)}{\sqrt{3}}\)
or
\(u=\frac{\sqrt{3}(i-j-k)}{3}\)

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