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

The vector projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{CD}}$, where $A \equiv(2,-3,0), B \equiv(1,-4,-2), C \equiv(4,6,8)$ and $\mathrm{D} \equiv(7,0,10)$, is
  1. $\frac{1}{49}(3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$
  2. $\frac{1}{6}(-\hat{i}-\hat{j}-2 \hat{k})$
  3. $-\frac{1}{49}(3 \hat{i}-6 \hat{j}+2 \hat{k})$
  4. $-\frac{1}{6}(-\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$

Solution

$\begin{aligned} & \overline{\mathrm{AB}}=-\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}} \\ & \overline{\mathrm{CD}}=3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}} \end{aligned}$ Vector projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{CD}}$ $\begin{aligned} & =(\overline{\mathrm{AB}} \cdot \overline{\mathrm{CD}}) \frac{\overline{\mathrm{CD}}}{|\overline{\mathrm{CD}}|^2} \\ & =(-3+6-4) \frac{(3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})}{\left(\sqrt{3^2+(-6)^2+2^2}\right)^2} \\ & =\frac{-1}{49}(3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \end{aligned}$

Asked in: MHT CET 2023 (10 May Shift 1)

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