Question

已知向量a,b的夹角为120°,且丨a丨=丨b丨=4,那么b*(2a+b)=?丨a+b丨=?,丨2a-b)=?

Answer 1

向量b·(2a+b)=b·2a+b·b

=2|a|*|b|*cos120°+|b|^2

=2*4*4*(-1/2)+4^2

=-16+16

=0.

|a+b|^2=(a+b)^2=|a|^2+2a·b+|b|^2

=4^2+2*4*4*cos120°+4^2

=16+32*(-1/2)+16

=16,

∴|a+b|=4,

|2a-b|^2=4|a|^2-4a·b+|b|^2

=4*4^2-4|a|*|b|*cos120°+4^2

=64-4*4*4*(-1/2)+16

=80+32

=112.

∴|2a-b|=4√7.