设向量a=(1,-2,2),b=(-3,x,4)已知a在b上的投影是1,求x

问题描述:

设向量a=(1,-2,2),b=(-3,x,4)已知a在b上的投影是1,求x

a.b
= (1,-2,2).(-3,x,4)
= -3-2x+8
= 5 -2x
a.b = |a||b| cosθ
a在b上的投影 = |a|cosθ
|b| = √ (9+x^2+16) = √(25+x^2)
(5-2x)/ √(25+x^2) = 1
5-2x = √(25+x^2)
25 - 20x +4x^2 = 25+x^2
3x^2-20x =0
x(3x-20) = 0
x = 0 or 20/3