Question

Determine the unit vector parallel to the cross product of the vectors $\overrightarrow{A}$ = $3\hat{i}-5\hat{j}+10\hat{k}$ and $\overrightarrow{B}$ = $6\hat{i}+5\hat{j}+2\hat{k}$

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

The unit vector parallel to $(\overrightarrow{A}\times \overrightarrow{B})$ is given by $\hat{n}$ =$\frac{\overrightarrow{A}\times \overrightarrow{B}}{\overrightarrow{|}A\times \overrightarrow{B}|}$
So, let us first determine $\overrightarrow{A}\times \overrightarrow{B}$
Now,$\overrightarrow{A}\times \overrightarrow{B}$ =$(3\hat{i}-5\hat{j}+10\hat{k})\times (6\hat{i}+5\hat{j}+2\hat{k})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& -5& 10\\ 6& 5& 2\end{array}\right|=-60\hat{i}+54\hat{j}+45\hat{k}$
Magnitude:$|\overrightarrow{A}\times \overrightarrow{B}|=\sqrt{(-60{)}^2+(54{)}^2+(45{)}^2}=\sqrt{8541}$
So required unit vector:$\hat{n}$ $\frac{-60\hat{i}+54\hat{j}+45\hat{k}}{\sqrt{8541}}$