#line geometry

Vectors and 3D Geometry

If the lines r = a_1 + b_1 ;and; r = a_2 + b_2 are coplanar then which of the following is correct ?

The trick is complete solution: write formula, plug data, do arithmetic. Final value matches ( a_2 - a_1 ) ( b_1 ; b_2 ) = 0. Simple and accurate.

Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry

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Vectors and 3D Geometry

What are the d.c's of the line whose d.r's are 1. -2. and 3 ?

The logic is straightforward: we verify 1 14, 2 14, 3 14 by full calculation: use standard identity, substitute, simplify step by step to reach 1 14, 2 14, 3 14.

Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry

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Vectors and 3D Geometry

If the direction cosines of a line are (1/k, 2/k, -2/k) then k is

Let's break it down: after simplifying, ± 3 is left. The method is simple: formula → substitution → calculation → answer.

Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry

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Vectors and 3D Geometry

If r = ​​ A_1 + B_1 and r = ​​ A_2 + B_2 are intersecting lines then which of the following is correct?

We know that after simplifying, 1 and 2 both is left. The method is simple: formula → substitution → calculation → answer.

Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry

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