First slide

Direction ratios and direction cosines

Question

$A (1, 2, 3), B (4, 5, 7), C (- 4, 3, - 6), D (2, k^{2}, 2)$ are four points. If the lines $\vec{A B} and \vec{C D}$ are parallel, then $k =$

Easy

A

2
B

6
C

3
D

None

Solution

If two lines are parallel, then their direction ratios are in proportional
The direction ration of line joining $A (x_{1}, y_{1}, z_{1}) and B (x_{2}, y_{2}, z_{2})$ is $〈x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}〉$
The direction ration of line joining $A (1, 2, 3) and B (4, 5, 7)$ is $〈3, 3, 4〉$
The direction ration of line joining $C (- 4, 3, - 6) and D (2, k^{2}, 2)$ is $〈6, k^{2} - 3, 8〉$
Since the lines are parallel, their direction ratios are proportional. It implies that $\frac{6}{3} = \frac{k^{2} - 3}{3} = \frac{8}{4}$
Therefore, $k^{2} - 3 = 6 \Rightarrow k^{2} = 9 \Rightarrow k = \pm 3$

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