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  4. Unit vector perpendicular to vector overset arrow A=-3 hati-2 hatj-3 hatk and overset arrow B=2 hati+4 hatj+6 hatk both is-

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Q. Unit vector perpendicular to vector $\overset{ \rightarrow }{A}=-3\hat{i}-2\hat{j}-3\hat{k}$ and $\overset{ \rightarrow }{B}=2\hat{i}+4\hat{j}+6\hat{k}$ both is-

NTA AbhyasNTA Abhyas 2022

Solution:

$\overset{ \rightarrow }{A}\times \overset{ \rightarrow }{B}$ lies perpendicular to the plane having $\overset{ \rightarrow }{A}$ and $\overset{ \rightarrow }{B}$ .
Thus,
$\overset{ \rightarrow }{A}\times \overset{ \rightarrow }{B}=\begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ -3 & -2 & -3 \\ 2 & 4 & 6 \end{bmatrix}$
$\Rightarrow \overset{ \rightarrow }{A}\times \overset{ \rightarrow }{B}=0\hat{i}+12\hat{j}-8\hat{k}$
$\Rightarrow \overset{ \rightarrow }{A}\times \overset{ \rightarrow }{B}=12\hat{j}-8\hat{k}$ .
The unit vector of $\overset{ \rightarrow }{A}\times \overset{ \rightarrow }{B}$ is
$\Rightarrow \frac{\overset{ \rightarrow }{A} \times \overset{ \rightarrow }{B}}{\left|\overset{ \rightarrow }{A} \times \overset{ \rightarrow }{B}\right|}=\frac{12 \hat{j} - 8 \hat{k}}{\sqrt{208}}$
$\Rightarrow \frac{\overset{ \rightarrow }{A} \times \overset{ \rightarrow }{B}}{\left|\overset{ \rightarrow }{A} \times \overset{ \rightarrow }{B}\right|}=\frac{12 \hat{j} - 8 \hat{k}}{4 \sqrt{13}}$
$\Rightarrow \frac{\overset{ \rightarrow }{A} \times \overset{ \rightarrow }{B}}{\left|\overset{ \rightarrow }{A} \times \overset{ \rightarrow }{B}\right|}=\frac{3 \hat{j} - 2 \hat{k}}{\sqrt{13}}$
Therefore, the unit vector perpendicular to vector $\overset{ \rightarrow }{A}=-3\hat{i}-2\hat{j}-3\hat{k}$ and $\overset{ \rightarrow }{B}=2\hat{i}+4\hat{j}+6\hat{k}$ both is $\frac{3 \hat{j} - 2 \hat{k}}{\sqrt{13}}$ .