Projection of one vector on another: Class 12 practice
The scalar projection of a on b is (a · b)/|b|. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Dot product
The idea behind projection of one vector on another
The scalar projection of a on b is (a · b)/|b|. The angle between them satisfies cos θ = (a · b)/(|a||b|). Perpendicular vectors have a · b = 0.
Step-by-step method
- Compute a · b = a₁b₁ + a₂b₂ + a₃b₃.
- Find |b| = √(b₁² + b₂² + b₃²).
- Projection = a · b ÷ |b|.
Worked example 1 (Basic)
Find a · b for a = (-5, -2, -2) and b = (4, 4, 7).
- -5×4 + -2×4 + -2×7
- Answer: -42
Worked example 2 (Standard)
Find the projection of a = (-3, 4, 1) on b = (2, 3, 6).
- a · b = 12, |b| = 7
- 12/7
- Answer: 12/7
Worked example 3 (Challenge)
Is (-8, 2, 0) perpendicular to (1, 4, 8)? Find the angle between them.
- Dot product = 0
- Answer: Yes, 90°
Common mistakes and misconceptions
- Dividing by |a|
- Forgetting the square root
Practice questions
- (Basic) Find a · b for a = (-2, -4, 5) and b = (4, 4, 7).
- (Basic) Find a · b for a = (4, -2, 3) and b = (2, 6, 9).
- (Standard) Find the projection of a = (-2, -3, -5) on b = (2, 3, 6).
- (Standard) Find the projection of a = (1, -5, -2) on b = (1, 2, 2).
- (Challenge) Is (-12, 8, 0) perpendicular to (2, 3, 6)? Find the angle between them.
- (Challenge) Is (-6, 2, 0) perpendicular to (2, 6, 9)? Find the angle between them.
Answer key with full working
- Q1: -2×4 + -4×4 + 5×7 → Answer: 11
- Q2: 4×2 + -2×6 + 3×9 → Answer: 23
- Q3: a · b = -43, |b| = 7 → -43/7 → Answer: -43/7
- Q4: a · b = -13, |b| = 3 → -13/3 → Answer: -13/3
- Q5: Dot product = 0 → Answer: Yes, 90°
- Q6: Dot product = 0 → Answer: Yes, 90°
Syllabus note
This skill is listed under "Vector Algebra" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is projection of one vector on another in?
"Vector Algebra", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.