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 $$i+k$$.

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
$$u=\frac{\sqrt{3}(i-j-k)}{3}$$