CBSE Class 12 Mathematics: Vectors and 3D Geometry — Important Questions with Answers 2026

Short Answer Questions (2-3 Marks)

• Q: Find the angle between vectors a⃗ = 2î + 3ĵ + k̂ and b⃗ = î - 2ĵ + 3k̂.
  Ans: a⃗·b⃗ = 2(1) + 3(-2) + 1(3) = 2 - 6 + 3 = -1. |a⃗| = √(4+9+1) = √14. |b⃗| = √(1+4+9) = √14. cos θ = (a⃗·b⃗)/(|a⃗||b⃗|) = -1/14. θ = cos⁻¹(-1/14).
• Q: Find the shortest distance between lines r⃗ = î + ĵ + λ(2î - ĵ + k̂) and r⃗ = 2î + ĵ - k̂ + μ(3î - 5ĵ + 2k̂).
  Ans: a₁ = î + ĵ, b₁ = 2î - ĵ + k̂. a₂ = 2î + ĵ - k̂, b₂ = 3î - 5ĵ + 2k̂. a₂ - a₁ = î - k̂. b₁ × b₂ = |î ĵ k̂; 2 -1 1; 3 -5 2| = î(-2+5) - ĵ(4-3) + k̂(-10+3) = 3î - ĵ - 7k̂. |b₁ × b₂| = √(9+1+49) = √59. d = |(a₂-a₁)·(b₁×b₂)|/|b₁×b₂| = |3-0+7|/√59 = 10/√59 units.
• Q: Find the equation of the plane passing through (1, 2, 3) and perpendicular to the vector 2î + 3ĵ - k̂.
  Ans: The equation of a plane through point (x₁,y₁,z₁) with normal vector (a,b,c) is: a(x-x₁) + b(y-y₁) + c(z-z₁) = 0. Here: 2(x-1) + 3(y-2) + (-1)(z-3) = 0. 2x - 2 + 3y - 6 - z + 3 = 0. 2x + 3y - z = 5.

Exam Tips for This Chapter

• Revise all definitions and laws from Vectors and 3D Geometry — commonly asked as 1-2 mark questions
• Practice diagrams related to Vectors and 3D Geometry — neat labelled diagrams carry 2-3 marks
• For numericals, always show formula → substitution → answer with correct units
• Previous year analysis shows Vectors and 3D Geometry carries significant marks in the board exam