Class 12 Vector Algebra · Short Answer2 Marksvector_identities
Class 12 Maths · Vector Algebra · Short Answer #58
Prove |a × b|² = |a|²|b|² − (a · b)² for any vectors a,b.
✅ Step-by-Step Solution
Final Answer: |a × b|² = |a|²|b|² − (a · b)².
- If either vector is zero, both sides vanish. Otherwise let θ be their angle.
- Use a · b = |a||b|cos θ and |a × b| = |a||b|sin θ. Then |a|²|b|² − (a · b)² = |a|²|b|²(1 − cos²θ) = |a|²|b|²sin²θ.
- This equals |a × b|², completing the proof including zero-vector cases.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] If either vector is zero, both sides vanish. Otherwise let θ be their angle.
- [1 mark] Use a · b = |a||b|cos θ and |a × b| = |a||b|sin θ. Then |a|²|b|² − (a · b)² = |a|²|b|²(1 − cos²θ) = |a|²|b|²sin²θ.
- [0.5 marks] This equals |a × b|², completing the proof including zero-vector cases.
Master Chapter 10: Vector Algebra
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Vector Algebra.
← Previous Question
For a = (1,2,−3), b = (3,−1,2), show a + b is perpendicular to a − b.
Next Question →If |a| = 3, |b| = 2/3 and a × b is a unit vector, find all possible…
More Practice Questions in Vector Algebra
Related Questions in Vector Algebra
Class 12 Vector Algebra · Mcq Single Correct
Find |a × b|² + (a · b)².
Class 12 Vector Algebra · Mcq Single Correct
Find the area of triangle OAB, where OA = (1,2,3) and OB = (−3,−2,1).
Class 12 Vector Algebra · Mcq Single Correct
For nonzero vectors a,b with angle θ ∈ [0,π], when is a · b ≥ 0?
Class 12 Vector Algebra · Mcq Single Correct
For a vector a, find (a · i)i + (a · j)j + (a · k)k.
Class 12 Vector Algebra · Mcq Single Correct
Which vector has magnitude 9 and the same direction as (1,−2,2)?
Class 12 Vector Algebra · Mcq Single Correct
Find the position vector dividing the segment from 2a − 3b to a + b internally in the ratio…