First slide

Direction ratios and direction cosines

Question

If $A (1, 2, 3), B (2, 3, 4), C (3, 4, 5), D (4, 5, 6)$ are four points, then the projection of $\bar{B D} on \bar{A C}$ is

Moderate

A

$\sqrt{3}$
B

$2 \sqrt{3}$
C

6
D

$\sqrt{6}$

Solution

The projection of $\bar{B D} on \bar{A C}$ is $\frac{\bar{B D} \cdot \bar{A C}}{|\bar{A C}|}$
Hence,
$\begin{matrix} \frac{\bar{B D} \cdot \bar{A C}}{|\bar{A C}|} = \frac{(\bar{O B} - \bar{O D}) \cdot (\bar{O C} - \bar{O A})}{|(\bar{O C} - \bar{O A})|} \\ = |\frac{[(2 i + 3 j + 4 k) - (4 i + 5 j + 6 k)] \cdot [(3 i + 4 j + 5 k) - (i + 2 j + 3 k)]}{|[(3 i + 4 j + 5 k) - (i + 2 j + 3 k)]|}| \\ = |\frac{(- 2 i - 2 j - 2 k) \cdot (2 i + 2 j + 2 k)}{\sqrt{4 + 4 + 4}}| \\ = \frac{12}{\sqrt{12}} \end{matrix}$
Therefore, the required projection is $2 \sqrt{3}$

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