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If $\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$, then the projection of $\bar{b}$ in the direction of $\bar{a}$ is
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Verified Answer
The correct answer is:
$\frac{3}{\sqrt{29}}$
Projection of $\overline{\mathrm{b}}$ in the direction of $\overline{\mathrm{a}}=\frac{\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}}{|\overline{\mathrm{a}}|}$
$$
\begin{aligned}
& =\frac{(2 \hat{i}+3 \hat{j}-4 \hat{k}) \cdot(\hat{i}-\hat{j}-\hat{k})}{\sqrt{2^2+3^2+(-4)^2}} \\
& =\frac{2-3+4}{\sqrt{4+9+16}}=\frac{3}{\sqrt{29}}
\end{aligned}
$$
$$
\begin{aligned}
& =\frac{(2 \hat{i}+3 \hat{j}-4 \hat{k}) \cdot(\hat{i}-\hat{j}-\hat{k})}{\sqrt{2^2+3^2+(-4)^2}} \\
& =\frac{2-3+4}{\sqrt{4+9+16}}=\frac{3}{\sqrt{29}}
\end{aligned}
$$
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