若e1,e2为夹角为60度的单位向量,则a=2e1+e2,b=-3e1+2e2的夹角为?

Question

若e1,e2为夹角为60度的单位向量,则a=2e1+e2,b=-3e1+2e2的夹角为?

Answer 1

a=2e1+e2 |a|=|2e1+e2|
|a|^2=|2e1+e2|^2=4e1^2+4e1e2+e2^2=4+4*|e1|*|e2|*cos60°+1=7
b=-3e1+2e2 |b|=|-3e1+2e2|
|b|^2=|-3e1+2e2|^2=9e1^2-12e1e2+4e2^2=13-12*|e1|*|e2|*cos60°=7
|a|=|b|=√7
e1+e2=|e1|*|e2|*cos60°=1/2
cos=a*b/|a|*|b|=(2e1+e2)*(-3e1+2e2)/7=(-6+2+e1*e2)/7
=-1/2
=120°