Question

Let vector A=a1i +a2j + a3k and B=b1i + b2j + b3k ne on the same plane. Fine the unit vector perpendicular to both A and B.

Answer 1

$$A\times B=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$

$$=(a_2b_3-a_3 b_2)\vec{i}+(a_3b_1-a_1b_3)\vec{j}+(a_1b_2-a_2b_1)\vec{k}$$

$$\vec {u}=\langle\dfrac{a_2b_3-a_3 b_2}{\sqrt{(a_2b_3-a_3 b_2)^2+(a_3b_1-a_1 b_3)^2+(a_1b_2-a_2 b_1)^2}},$$

$$\dfrac{a_3b_1-a_1b_3}{\sqrt{(a_2b_3-a_3 b_2)^2+(a_3b_1-a_1 b_3)^2+(a_1b_2-a_2 b_1)^2}},$$

$$\dfrac{a_1b_2-a_2b_1}{\sqrt{(a_2b_3-a_3 b_2)^2+(a_3b_1-a_1 b_3)^2+(a_1b_2-a_2 b_1)^2}}\rangle$$