If ( a⃗ + 3b⃗ ) is perpendicular to ( 7a⃗ - 5b⃗ ) and ( a⃗ - 4b⃗ ) is perpendicular to ( 7a⃗ - 2b⃗ ), then the angle between a⃗ and b⃗ (in degrees) is _______________.

Solution

Maths sir · 4 steps

Turn each perpendicular pair into a dot product equal to zero, then solve two equations together.

1. Perpendicular means zero. Two vectors are perpendicular when their dot product is zero. Expand each product using a⃗ · b⃗ = b⃗ · a⃗, a⃗ · a⃗ = |a⃗|² and b⃗ · b⃗ = |b⃗|².
2. Two equations. First pair: 7|a⃗|² - 5 a⃗·b⃗ + 21 a⃗·b⃗ - 15|b⃗|² = 0, so 7|a⃗|² + 16 a⃗·b⃗ - 15|b⃗|² = 0. Second pair: 7|a⃗|² - 2 a⃗·b⃗ - 28 a⃗·b⃗ + 8|b⃗|² = 0, so 7|a⃗|² - 30 a⃗·b⃗ + 8|b⃗|² = 0.
3. Eliminate. Subtract the second equation from the first: 46 a⃗·b⃗ - 23|b⃗|² = 0, so a⃗·b⃗ = 1/2|b⃗|². Put this into the second equation: 7|a⃗|² - 15|b⃗|² + 8|b⃗|² = 0, so |a⃗| = |b⃗|.

   cos θ = (a⃗·b⃗)/(|a⃗| |b⃗|) = (1/2|b⃗|²)/|b⃗|² = 1/2

4. So. cos θ = 1/2 gives θ = 60 degrees.

The idea to keep. When a question gives you perpendicular combinations of two vectors, treat |a⃗|², |b⃗|² and a⃗·b⃗ as unknowns. Eliminate to get their ratios, and the angle follows.

Answer from the NTA final answer key. Solution by PrepWiser.