Question

己知ab是两个非零向量,且|a|=|b|=|a-b|,求向量a与a+b的夹角

Answer 1

设|a|=|b|=|a-b|=m

|a-b|²=a²-2a.b+b²

m²=2m²-2a.b

a.b=m²/2

a.(a+b)=a²+a.b=m²+m²/2=3m²/2

|a+b|²=a²+2a.b+b²=3m²

|a+b|=√3m

cos=[a.(a+b)]/[|a|*|a+b|]=(3m²/2)/[m(√3m)]=√3/2

所以,夹角是30°