Q28
jee_main_2024_31_jan_morning
Vector Triple Product
Let a and b be two vectors such that | a| = 1, | b| = 4 and a · b = 2. If c = (2 a × b) - 3 b and the angle between b and c is α, then 192 ²α is equal to
Numerical Answer. Answer: 48 to 48
Solution
Core Logic
b · c = b · ((2 a × b) - 3 b) |b||c| α = 2( b · ( a × b)) - 3|b|²Since b · ( a × b) = 0, we have |b||c| α = -3|b|².
|c| α = -3|b| = -12 |c|² ² α = 144Step 1: Compute Modulus of c
|c|² = |2 a × b - 3 b|² = 4| a × b|² + 9| b|² - 12(( a × b) · b) = 4| a × b|² + 9| b|²Given a · b = 2 |a||b| θ = 2 1 · 4 θ = 2 θ = (π)/(3).
| a × b|² = |a|²|b|² ²θ = 1 · 16 · (3)/(4) = 12 |c|² = 4(12) + 9(16) = 48 + 144 = 192Step 2: Final Calculation
We know |c|² ² α = 144.
192 ² α = 144 192(1 - ² α) = 144 192 ² α = 192 - 144 = 48Chapter Mix
Class 12 Maths: Vector Algebra
