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A unit vector perpendicular to each of the vectors
$2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ is
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$2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ is
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Verified Answer
The correct answer is:
$\frac{1}{\sqrt{3}} \hat{\mathrm{i}}+\frac{1}{\sqrt{3}} \hat{\mathrm{j}}-\frac{1}{\sqrt{3}} \hat{\mathrm{k}}$
$\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 1 \\ 3 & -4 & -1\end{array}\right|$
$=\hat{i}(1+4)-\hat{j}(-2-3)+\hat{k}(-8+3)$
$=5 \hat{i}+5 \hat{j}-5 \hat{k}$
Unit vector $=\frac{5}{5 \sqrt{3}} \hat{i}+\frac{5}{5 \sqrt{3}} \hat{j}-\frac{5}{5 \sqrt{3}} \hat{k}$
$\quad=\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}-\frac{1}{\sqrt{3}} \hat{k}$
$=\hat{i}(1+4)-\hat{j}(-2-3)+\hat{k}(-8+3)$
$=5 \hat{i}+5 \hat{j}-5 \hat{k}$
Unit vector $=\frac{5}{5 \sqrt{3}} \hat{i}+\frac{5}{5 \sqrt{3}} \hat{j}-\frac{5}{5 \sqrt{3}} \hat{k}$
$\quad=\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}-\frac{1}{\sqrt{3}} \hat{k}$
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