The direction cosines of the line passing through the points and are:
The direction cosines of the line passing through the points $P(2, 3, 5)$ and $Q(-1, 2, 4)$ are:
- a$\left(\frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{1}{\sqrt{11}}\right)$
- b$\left(\frac{1}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{1}{\sqrt{11}}\right)$
- c$\left(\frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}}, \frac{-1}{\sqrt{11}}\right)$
- d$\left(\frac{-3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{-1}{\sqrt{11}}\right)$
Answer
Answer
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Correct option: a
Answer: (a) $\left(\frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{1}{\sqrt{11}}\right)$
Direction ratios of the line from $Q(-1, 2, 4)$ to $P(2, 3, 5)$ are $(2 - (-1), 3 - 2, 5 - 4) = (3, 1, 1)$. The magnitude is $\sqrt{3^2 + 1^2 + 1^2} = \sqrt{11}$, giving direction cosines $\left(\frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{1}{\sqrt{11}}\right)$.
From ISC 2026 Mathematics Paper 1, question 15(iii).