TS EAMCET · Maths · Three Dimensional Geometry

The direction cosines of the line passing through $P (2, 3, - 1)$ and the origin are

A $\frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}$
B $\frac{2}{\sqrt{14}}, \frac{- 3}{\sqrt{14}}, \frac{1}{\sqrt{14}}$
C $\frac{- 2}{\sqrt{14}}, \frac{- 3}{\sqrt{14}}, \frac{1}{\sqrt{14}}$
D $\frac{2}{\sqrt{14}}, \frac{- 3}{\sqrt{14}}, \frac{- 1}{\sqrt{14}}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{- 2}{\sqrt{14}}, \frac{- 3}{\sqrt{14}}, \frac{1}{\sqrt{14}}$

Step-by-step Solution

Detailed explanation

Given points are p2,3,-1=x1,y1,z1 and 0,0,0=x2,y2,z2 Direction cosines of a line passing through these two points is given by cosα=x2-x1x12+y12+z12 cosβ=y2-y1x12+y12+z12 cosγ=z2-z1x12+y12+z12 cosα=0-222+32+-12=-214 cosβ=0-322+32+-12=-314 …

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The direction cosines of the line passing through P2, 3, -1 and the origin are

Step-by-step Solution

The direction cosines of the line passing through $P (2, 3, - 1)$ and the origin are