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

Solution:

Given, a = i + j = (1, 1, 0)

b = i + k = (1, 0, 1)

We have to find the two unit vectors orthogonal to both a and b.

Finding cross product,

\(a\times b=\begin{vmatrix} i &j &k \\ 1 &1 &0 \\ 1 & 0 &1 \end{vmatrix}\)

\(\\=i(1-0)-j(1-0)+k(0-1)\\=i-j-k\)

= (1, -1, -1)

\(\left | \vec{a}\times \vec{b} \right |=\sqrt{(1)^{2}+(-1)^{2}+(-1)^{2}}=\sqrt{1+1+1}=\sqrt{3}\)

Now, unit vector perpendicular to \(\vec{a}\, and\, \vec{b}=\pm \frac{(\vec{a}\times \vec{b})}{\left | \vec{a}\times \vec{b} \right |}\)

= \(\pm \frac{1}{\sqrt{3}}(1, -1, -1)\)

Therefore, the unit vectors are \(\pm \frac{1}{\sqrt{3}}(1, -1, -1)\)

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

Summary:

The unit vector orthogonal to both i + j and i + k are \(\pm \frac{1}{\sqrt{3}}(1, -1, -1)\).