Vector Algebra — Class 12 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Vector Algebra" — 8 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Vector Algebra" — 8 important questions with detailed answers for CBSE board exam…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

Find the magnitude of the vector a = 3î + 4ĵ.
Find the dot product of vectors a = 2î + 3ĵ + k̂ and b = î + 2ĵ - k̂.
Find the cross product a × b where a = î + 2ĵ + 3k̂ and b = 2î + 3ĵ + k̂.
Find the angle between vectors a = î + ĵ and b = ĵ + k̂.
Find the scalar projection of a = 2î + 3ĵ - k̂ on b = î + ĵ + k̂.
Verify that vectors a = 2î + 3ĵ and b = -3î + 2ĵ are perpendicular.
+ 2 more questions in the full chapter

Solutions Summary:

Question	Status
Find the magnitude of the vector a = 3î + 4ĵ.	✓ Solved
Find the dot product of vectors a = 2î + 3ĵ + k̂ and b = …	✓ Solved
Find the cross product a × b where a = î + 2ĵ + 3k̂ and b…	✓ Solved
Find the angle between vectors a = î + ĵ and b = ĵ + k̂.	✓ Solved
Find the scalar projection of a = 2î + 3ĵ - k̂ on b = î +…	✓ Solved
Verify that vectors a = 2î + 3ĵ and b = -3î + 2ĵ are perp…	✓ Solved

Showing 6 of 8 questions

Q1: Find the magnitude of the vector a = 3î + 4ĵ.

To find the magnitude of a = 3î + 4ĵ: Step 1: Use the magnitude formula. For a vector a = xî + yĵ + zk̂, |a| = √(x² + y² + z²) Step 2: Identify components. Here: x = 3, y = 4, z = 0 Step 3: Calculate. |a| = √(3² + 4² + 0²) = √(9 + 16) = √25 = 5 Final Answer: |a| = 5 units

Q2: Find the dot product of vectors a = 2î + 3ĵ + k̂ and b = î + 2ĵ - k̂.

To find a · b where a = 2î + 3ĵ + k̂ and b = î + 2ĵ - k̂: Step 1: Use dot product formula. a · b = (a₁î + a₂ĵ + a₃k̂) · (b₁î + b₂ĵ + b₃k̂) = a₁b₁ + a₂b₂ + a₃b₃ Step 2: Identify components. a₁ = 2, a₂ = 3, a₃ = 1 b₁ = 1, b₂ = 2, b₃ = -1 Step 3: Calculate. a · b = (2)(1) + (3)(2) + (1)(-1) = 2 + 6 - 1 = 7 Final Answer: a · b = 7

Q3: Find the cross product a × b where a = î + 2ĵ + 3k̂ and b = 2î + 3ĵ + k̂.

To find a × b where a = î + 2ĵ + 3k̂ and b = 2î + 3ĵ + k̂: Step 1: Use the determinant formula. a × b = |î ĵ k̂ | |1 2 3 | |2 3 1 | Step 2: Expand the determinant. a × b = î(2×1 - 3×3) - ĵ(1×1 - 3×2) + k̂(1×3 - 2×2) = î(2 - 9) - ĵ(1 - 6) + k̂(3 - 4) = î(-7) - ĵ(-5) + k̂(-1) = -7î + 5ĵ - k̂ Final Answer: a × b = -7î + 5ĵ - k̂

Q4: Find the angle between vectors a = î + ĵ and b = ĵ + k̂.

To find the angle θ between a = î + ĵ and b = ĵ + k̂: Step 1: Calculate the dot product a · b. a · b = (1)(0) + (1)(1) + (0)(1) = 1 Step 2: Calculate magnitudes. |a| = √(1² + 1² + 0²) = √2 |b| = √(0² + 1² + 1²) = √2 Step 3: Use the formula cos θ = (a · b)/(|a||b|). cos θ = 1/(√2 × √2) = 1/2 Step 4: Find θ. θ = cos⁻¹(1/2) = π/3 or 60° Final Answer: θ = π/3 (or 60°)

Q5: Find the scalar projection of a = 2î + 3ĵ - k̂ on b = î + ĵ + k̂.

To find scalar projection of a on b: Step 1: Use scalar projection formula. Scalar projection of a on b = (a · b)/|b| Step 2: Calculate a · b. a · b = (2)(1) + (3)(1) + (-1)(1) = 2 + 3 - 1 = 4 Step 3: Calculate |b|. |b| = √(1² + 1² + 1²) = √3 Step 4: Calculate scalar projection. Scalar projection = 4/√3 = 4√3/3 Final Answer: Scalar projection = 4/√3 = 4√3/3

Q6: Verify that vectors a = 2î + 3ĵ and b = -3î + 2ĵ are perpendicular.

To verify that a and b are perpendicular: Step 1: Recall perpendicularity condition. Two vectors are perpendicular if and only if a · b = 0 Step 2: Calculate the dot product. a · b = (2)(-3) + (3)(2) + (0)(0) = -6 + 6 + 0 = 0 Step 3: Conclusion. Since a · b = 0, the vectors are perpendicular. Final Answer: Yes, vectors a and b are perpendicular (verified)

Showing 6 of 8 questions. Visit the full page for complete solutions.