If  A1,2,3,B2,3,4,C3,4,5,D4,5,6are four points, then the projection of BD¯  on  AC¯  is

# If  $A\left(1,2,3\right),B\left(2,3,4\right),C\left(3,4,5\right),D\left(4,5,6\right)$are four points, then the projection of $\overline{BD}\text{\hspace{0.17em}\hspace{0.17em}on\hspace{0.17em}\hspace{0.17em}}\overline{AC}$  is

1. A

$\sqrt{3}$

2. B

$2\sqrt{3}$

3. C

6

4. D

$\sqrt{6}$

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### Solution:

The projection of $\overline{BD}\text{\hspace{0.17em}\hspace{0.17em}on\hspace{0.17em}\hspace{0.17em}}\overline{AC}$ is$\frac{\overline{BD}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\overline{AC}}{\left|\overline{AC}\right|}$

Hence,

$\begin{array}{c}\frac{\overline{BD}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\overline{AC}}{\left|\overline{AC}\right|}=\frac{\left(\overline{OB}-\overline{OD}\right)\cdot \left(\overline{OC}-\overline{OA}\right)}{\left|\left(\overline{OC}-\overline{OA}\right)\right|}\\ =\left|\frac{\left[\left(2i+3j+4k\right)-\left(4i+5j+6k\right)\right]\cdot \left[\left(3i+4j+5k\right)-\left(i+2j+3k\right)\right]}{\left|\left[\left(3i+4j+5k\right)-\left(i+2j+3k\right)\right]\right|}\right|\\ =\left|\frac{\left(-2i-2j-2k\right)\cdot \left(2i+2j+2k\right)}{\sqrt{4+4+4}}\right|\\ =\frac{12}{\sqrt{12}}\end{array}$

Therefore, the required projection is  $\overline{)2\sqrt{3}}$

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