Question

For $A(1,-2,4),B(5,-1,7),C(3,6,-2),D(4,5,-1)$, the projection of $\overline{AB}$ on $\overline{CD}$ is ________.

• A. $(2\sqrt{3},-2\sqrt{3},2\sqrt{3})$
• B. $\frac{3}{13}(4,1,3)$
• C. $(1,-1,1)$
• D. $(2,-2,2)$

Answer 1

Answer: (D)

$(2,-2,2)$

$\overline{AB}=\overline{B}-\overline{A}$

$A(1,-2,4),B(5,-1,7)$ ....given

$\overline{A}=\overline{i}-2\overline{j}+4\overline{k}$ and

$\overline{B}=5\overline{i}-\overline{j}+7\overline{k}$

$\therefore \overline{AB}=(5-1)\overline{i}+(-1+2)\overline{k}+(7-4)\overline{k}$

$\therefore \overline{AB}=4\overline{i}+\overline{j}+3\overline{k}$

$\overline{AB}=(4,1,3)$

$C(3,6,-2),D(4,5,-1)$

$\overline{C}=3\overline{i}+6\overline{j}-2\overline{k}$ and

$\overline{D}=4\overline{i}+5\overline{j}-\overline{k}$

$\therefore \overline{CD}=(4-3)\overline{i}+(5-6)\overline{k}+(-1+2)\overline{k}$

$\therefore \overline{CD}=\overline{i}+\overline{j}-\overline{k}$

$\overline{CD}=(1,-1,1)$

Now projection of $\overline{AB}$ on $\overline{C}D=$

$\left(\frac{\overline{AB}\cdot \overline{CD}}{|CD|}\right)\cdot \widehat{CD}$

$=\left(\frac{4-1+3}{\sqrt{4}}\right)\cdot \frac{(1,-1,1)}{\sqrt{3}}$

$=(2,-2,2)$.