Given the vectors ||a|| = 8, ||b|| = 12, and ||a+b|| = 16, find all angles between each other

Question

1. Given the vectors ||a|| = 8, ||b|| = 12, and ||a+b|| = 16, find all angles between each other

Answer 1

Ia + bI² = (a + b)² = a² + 2ab + b² = IaI² + 2IaI·IbIcosθ + IbI^2; 16² = 8² + 12² + 2·8·12cosθ;

256 - 64 - 144 = 192cosθ; cosθ = 48/192 = 0.25; θ = cos^-10.25 = 75.52°; m ∠(a, b) = 75.52°;

a(a + b) = IaI² + IaI·IbIcosθ = IaI·Ia + bIcosφ; 64 + 8·12·0.25 = 8 ·16cosφ; cosφ = 88/128 = 11/16 = 0.6875

φ = cos^-10.6875 = 46.57°; m ∠(a,a + b) = 46.57°

b(a + b) = IbI² + IaI·IbIcosθ = IbI·Ia + bIcos β; 144 + 8·12·0.25 = 12 ·16cosβ; cosβ =168/192 = 0.875;β

β = cos^-10.875 = 28.96°; m ∠(b,a + b) = 28.96°

φ is angle between a and (a + b)