若向量a与b夹角为30度,且|a|=根号3,|b|=1,则向量p=a+b与q=a-b的夹角余弦值为?

题目

若向量a与b夹角为30度,且|a|=根号3,|b|=1,则向量p=a+b与q=a-b的夹角余弦值为?

答案

由已知|a| = √3,|b| = 1,向量a与向量b的夹角为30°,所以|a + b| 2 = (a + b)·(a + b) = a 2 + 2a·b + b 2 = |a| 2 + 2*|a|*|b|*cos30° + |b| 2 = (√3) 2 + 2√3(√3/2) + 1 2 = 3 + 3 + 1 = 7,开方可得p = |a + b| = √7 ； 同理,|a – b| 2 = (a – b·(a – b) = a 2 –2a·b + b 2 = |a| 2 –2*|a|*|b|*cos30° + |b| 2 = (√3) 2 –2√3(√3/2) + 1 2 = 3 – 3 + 1 = 1,开方可得q = |a – b| = 1 ； 设向量p = a + b与q = a – b的夹角为θ,由点乘公式可得p·q = |p|*|q|*cosθ= (a + b)·(a – b) = a 2 –b 2 = |a| 2 –|b| 2 = (√3) 2 –1 2 = 3 – 1 = 2,化简可得√7*1*cosθ = 2,所以cosθ = 2/√7 = 2√7/7 ； 综上所述,向量p和向量q的夹角的余弦值为 2 √7/7 .