First slide

Direction ratios and direction cosines

Question

The square of the projection of the join of the points $(1, 4, 7)$ and $(2, 5, 8)$ on the line whose direction cosines are $(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$ , is

Easy

A

3
B

4
C

5
D

6

Solution

The length of the projection of the line segment joining $A (x_{1}, y_{1}, z_{1}), B (x_{2}, y_{2}, z_{2})$ on the line
whose direction cosines are $l, m, n$ is $l (x_{2} - x_{1}) + m (y_{2} - y_{1}) + n (z_{2} - z_{1})$
Hence, the projection of the join of the points $(1, 4, 7)$ and $(2, 5, 8)$ on the line
whose direction cosines are $(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$ is $\frac{1}{\sqrt{3}} (1 + 1 + 1) = \sqrt{3}$

Therefore, the required value is $3$

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