The projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{CD}}$, where $\mathrm{A} \equiv(2,-3,0),…

The projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{CD}}$, where $\mathrm{A} \equiv(2,-3,0), \mathrm{B} \equiv(1,-4,-2), \mathrm{C} \equiv(4,6,8)$ and $\mathrm{D} \equiv(7,0,10)$ is
  1. $\frac{1}{\sqrt{6}}$
  2. $\frac{1}{7}$
  3. $\frac{4}{\sqrt{6}}$
  4. $\frac{4}{7}$

Solution

$\begin{aligned} & \overline{\mathrm{AB}}=-\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}} \\ & \overline{\mathrm{CD}}=3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}} \end{aligned}$
Scalar projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{CD}}$ $=\frac{\overline{\mathrm{AB}} \cdot \overline{\mathrm{CD}}}{\overline{\mathrm{CD}}}=\frac{-3+6-4}{\sqrt{9+36+4}}=\frac{1}{7}$

Asked in: MHT CET 2024 (10 May Shift 2)

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