If 𝐚 and 𝐛 are two vectors such that |𝐚 + 𝐛| = |𝐚|, then prove that 2𝐚 + 𝐛 is perpendicular to 𝐛.

Given: |𝐚 + 𝐛| = |𝐚|
Squaring both sides, we get
|𝐚 + 𝐛|² = |𝐚|²
(𝐚 + 𝐛) • (𝐚 + 𝐛) = 𝐚 • 𝐚
|𝐚|² + 2𝐚 • 𝐛 + |𝐛|² = |𝐚|²
2𝐚 • 𝐛 + |𝐛|² = 0
2𝐚 • 𝐛 = -|𝐛|²
(2𝐚 + 𝐛) • 𝐛 = 2𝐚 • 𝐛 + 𝐛 • 𝐛 = 2𝐚 • 𝐛 + |𝐛|² = -|𝐛|² + |𝐛|² = 0
Since the dot product of (2𝐚 + 𝐛) and 𝐛 is 0, the vectors (2𝐚 + 𝐛) and 𝐛 are perpendicular.