Question

已知向量a=(4,-3),b=(2,1)若a+tb与b的夹角为45度,求实数t

Answer 1

a+tb=(4+
2t
,-3+t)
(a+tb)*b=2(4+2t)+(-3+t)=5+5t
又有(a+tb)*b=|a+tb||b|cos45
5+5t=根号[(4+2t)^2+(-3+t)^2]*根号(4+1)*根号2/2
25+50t+
25t
^2=(16+16t+
4t
^2+9-6t+t^2)*5/2
10+20t+10t^2=5t^2+10t+25
5t^2+10t-15=0
t^2+2t-3=0
(t+3)(t-1)=0
t=-3或t=1
t=-3带入a+tb与b的夹角是3π/4所以t=-3舍去
t=1

Answer 2

a+tb=（4+2t,t-3）
(a+tb)·b=|a+tb||b|cos45°
解出来就有t了