Find two unit vectors orthogonal to both 3, 5, 1 and -1, 1, 0 .

Solution:

Considering x = (3, 5, 1)

y = (-1, 1, 0)

\(\overline{x}\times \overline{y}=\begin{vmatrix} i & j & k\\ 3 & 5 & 1\\ -1 & 1 & 0 \end{vmatrix}\)

= i (0 - 1) - j (0 + 1) + k(3 + 5)

= -i - j + 8k

= (-1, -1, 8)

\(|\overline{x}\times \overline{y}|=\sqrt{1+1+64}=\sqrt{66}\)

Unit vector orthogonal to x and y 

\(\\=\pm \frac{(\overline{x}\times \overline{y})}{|\overline{x}\times \overline{y}|} \\ \\=\pm \frac{1}{\sqrt{66}}(-1,-1,8)\)

Therefore, two unit vectors are \(\pm \frac{1}{\sqrt{66}}(-1,-1,8)\)

Find two unit vectors orthogonal to both 3, 5, 1 and -1, 1, 0 .

Summary:

The two unit vectors orthogonal to both 3, 5, 1 and -1, 1, 0 are \(\pm \frac{1}{\sqrt{66}}(-1,-1,8)\)