Find two unit vectors orthogonal to both 4, 6, 1 and -1, 1, 0 .

Solution:

Let vector {a} = (4, 6, 1) = 4i + 6j + k

vector b = (-1, 1, 0) = -i + j

Two unit vectors orthogonal to both vector {a} and vector b

= ± [\(\vec{a}\) × \(\vec{b}]\) / ┃[\(\vec{a}\) × \(\vec{b}] \)┃

[\(\vec{a}\) × \(\vec{b}]\) = \(\begin{vmatrix} i & j & k\\ 4 &6 & 1 \\ -1 &1 &0 \end{vmatrix}\)

[\(\vec{a} \)× \(\vec{b}] \)= i (0 - 1) - j (0 + 1) + k (4 + 6)

[\(\vec{a}\) × \(\vec{b}]\) = -i - j + 10k

┃[\(\vec{a}\) × \(\vec{b}]\) ┃= √[(-1)² + (-1)² + 10²] = √102

Required vector = ± [- i - j + 10k] / √102

Find two unit vectors orthogonal to both 4, 6, 1 and -1, 1, 0 .

Summary:

Two unit vectors orthogonal to both 4, 6, 1 and -1, 1, 0 are ± [- i - j + 10k] / √102.