Dot product, angle and cross product: Class 12 practice
a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Magnitude of a vector
The idea behind dot product, angle and cross product
a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. a × b is perpendicular to both, with |a × b| = |a||b| sin θ = area of the parallelogram on a and b.
Step-by-step method
- Compute a·b component-wise.
- cos θ = a·b ÷ (|a||b|).
- For a × b use the determinant with i, j, k.
Worked example 1 (Basic)
Find a·b for a = 1i − 3j − 1k and b = 3i + 4j − 1k.
- 1×3 + -3×4 + -1×-1
- = -8
- Answer: -8
Worked example 2 (Standard)
Find a × b for a = 0i − 3j − 2k and b = -4i − 1j + 4k.
- i(-3·4 − -2·-1) − j(0·4 − -2·-4) + k(0·-1 − -3·-4)
- = -14i + 8j − 12k
- Answer: -14i + 8j − 12k
Worked example 3 (Challenge)
Find the area of the parallelogram with adjacent sides a = -3i − 2j + 3k and b = -1i − 4j − 1k.
- a × b = 14i − 6j + 10k
- Area = |a × b| = √332
- Answer: √332
Common mistakes and misconceptions
- Adding components for a·b
- Wrong sign on the j component of a × b
Practice questions
- (Basic) Find a·b for a = 4i − 2j + 4k and b = -4i + 1j − 3k.
- (Basic) Find a·b for a = 1i − 2j − 4k and b = -1i + 3j − 3k.
- (Standard) Find a × b for a = -4i − 3j − 3k and b = 3i − 4j − 2k.
- (Standard) Find a × b for a = 1i − 3j + 4k and b = 1i − 2j + 1k.
- (Challenge) Find the area of the parallelogram with adjacent sides a = -1i + 1j − 4k and b = 1i + 2j + 1k.
- (Challenge) Find the area of the parallelogram with adjacent sides a = -1i + 2j + 3k and b = 0i + 3j + 2k.
Answer key with full working
- Q1: 4×-4 + -2×1 + 4×-3 → = -30 → Answer: -30
- Q2: 1×-1 + -2×3 + -4×-3 → = 5 → Answer: 5
- Q3: i(-3·-2 − -3·-4) − j(-4·-2 − -3·3) + k(-4·-4 − -3·3) → = -6i − 17j + 25k → Answer: -6i − 17j + 25k
- Q4: i(-3·1 − 4·-2) − j(1·1 − 4·1) + k(1·-2 − -3·1) → = 5i + 3j + 1k → Answer: 5i + 3j + 1k
- Q5: a × b = 9i − 3j − 3k → Area = |a × b| = √99 → Answer: √99
- Q6: a × b = -5i + 2j − 3k → Area = |a × b| = √38 → Answer: √38
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 dot product, angle and cross product 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.