If $A, B, C$ and $D$ are points whose position vectors are…
If $A, B, C$ and $D$ are points whose position vectors are $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, 4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 5 \hat{\mathbf{i}}+\hat{\mathbf{j}}$, $7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ respectively, then the projection of $\mathbf{A B}$ on $\mathbf{C D}$ is
$\frac{4}{3}$
$\sqrt{\frac{2}{7}}$
$\frac{3}{4}$
$\sqrt{\frac{7}{2}}$
Solution
Given, position vectors of points $A, B, C$ and $D$ are $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, 4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 5 \hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$.
So, $\mathbf{A B}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{C D}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$
$\therefore$ Projection of $\mathbf{A B}$ on $\mathbf{C D}$ is
$
=\frac{|\mathbf{A B} \cdot \mathbf{C D}|}{|\mathbf{C D}|}=\frac{|6-2+3|}{\sqrt{4+1+9}}=\frac{7}{\sqrt{14}}=\sqrt{\frac{7}{2}}
$
Hence, option (d) is correct