已知向量a=3i+4j ,向量b=4i+3j ,向量c=m×向量a+n×向量b.且向量a⊥向量b,向量c的摩=1 求m,n的值

a=(3,4),b=(4,3),c=(3m+4n,4m+3n)
向量a⊥向量c
a.c=9m+12n+16m+12n=25m+24n=0
向量c的摩=1
|c|^2=1=(3m+4n)^2+(4m+3n)^2
=9mm+16nn+24mn+16mm+9nn+24mn
=m(25m+24n)+25nn+24mn
=25nn-(24*24nn/25)=1
nn=25/49
得n=5/7,m=-24/35或者n=-5/7,m=24/35