Question

为什么cosθ=a*b/|a||b| a,b均为向量 θ为a、b向量夹角

Answer 1

在直角坐标系xoy中,设A(x,y),B(m,n)两点,向量a=OA=(x,y) b=OB(m,n);
k1=y/x k2=n/m
tana=(k1-k2)/(1+k1*k2)=k =>k=(ym-nx)/(xm+ny)
cosa=1/(1+k^2)^0.5=(xm+ny)/((xn-ym)^2+(xm+ny)^2)^0.5
=(xm+yn)/((xn)^2+(ym)^2+(xm)^2+(yn)^2)^0.5
a*b=x*m+y*n
|a|*|b|=(x^2+y^2)^0.5*(m^2+n^2)^0.5
=>(a*b)/(|a|*|b|)=(xm+ny)/(x^2+y^2)^0.5*(m^2+n^2)^0.5
=(xm+yn)/((xn)^2+(ym)^2+(xm)^2+(yn)^2)^0.5
=>cosa=a*b/|a||b|