|a + tb|² = (a + tb)(a + tb)
= |a|² + t²|b|² + 2t(a • b)
= |a|² + t²|b|² + 2t * |a||b|cosθ
= |b|²t² + 2(|b|t)(|a|cosθ) + (|a|cosθ)² - (|a|cosθ)² + |a|²
= (|b|t + |a|cosθ)² + |a|²(1 - cos²θ)
= |b|²(t + |a|/|b| * cosθ)² + (|a|sinθ)²
當(dāng)t = - |a|/|b| * cosθ = - (a • b)/|b|² 時取得最小值
由t = - (a • b)/|b|² 得 |b|² = - (a • b)/t,用以代入下面的算式
∵b • (a + tb) = a • b + t|b|² = a • b + t * [- (a • b)/t] = a • b - a • b = 0
∴b⊥(a + tb),
即向量b垂直于向量u.