The Vectors and 3D Geometry category gathers questions that test understanding of vector fundamentals and their applications in three‑dimensional space. Topics include vector addition, scalar multiplication, dot and cross products, as well as lines, planes, distances and angles in 3‑D. Use these problems to strengthen spatial reasoning for physics and mathematics exams.
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
From the definition, using formula we get 〈3, 2, 1〉. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get 〈3, 2, 1〉.
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
Let's break it down: complete solution: write formula, plug data, do arithmetic. Final value matches plane parallel to xy - plane. Simple and accurate.
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
Understanding this is easy: we verify ^ - 1( 8/21 ) by full calculation: use standard identity, substitute, simplify step by step to reach ^ - 1( 8/21 ).
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
We know that using formula we get 2. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get 2.
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
Here's the idea in simple terms: we verify 5 ∶ 3 externally by full calculation: use standard identity, substitute, simplify step by step to reach 5 ∶ 3 externally.
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
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
Applying the basic concept, complete solution: write formula, plug data, do arithmetic. Final value matches y = 1. Simple and accurate.
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
Let's break it down: we verify π / 3 by full calculation: use standard identity, substitute, simplify step by step to reach π / 3.
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
In easy words, complete solution: write formula, plug data, do arithmetic. Final value matches 1. Simple and accurate.
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
Recall the rule solving fully: apply relevant formula, put numbers, compute. Result comes as ^ - 1( 5 58 ). Easy verification shows it satisfies condition.
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
The trick is using formula we get 1 3 ; i + 1 3 ; j + 1 3 ; k. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get 1 3 ; i + 1
Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry
