数学++++++++++++++++++++++已知向量a=(1,1),b(0,-2),尅为实数.(1)若向量ka-b与a+b共线,求实数k;(2)若向量ka-b与a+b的夹角为120°,求实数k.
2019-04-13
数学++++++++++++++++++++++
已知向量a=(1,1),b(0,-2),尅为实数.
(1)若向量ka-b与a+b共线,求实数k;
(2)若向量ka-b与a+b的夹角为120°,求实数k.
优质解答
一,ka-b为(k,k+2)a+b为(1,-2)共线k:1=k+2:-2得到k=-2/3
二,a² = 1 + 1 = 2
b² = 0 + 4 = 4
a · b = 0 - 2 = - 2
|ka-b|=|(k,k+2)|=√[k²+(k+2)²]=√(2k²+4k+4)
|a+b|=|(1,-1)|=√2
θ=120度
(ka - b) · (a + b)
= ka² + ka·b - b·a - b²
= ka² + (k-1) a·b - b²
= 2k - 2(k-1) - 4
= - 2
= |ka-b|·|a+b|·cosθ
整理得 (k+1)²=3
k= -1±√3
一,ka-b为(k,k+2)a+b为(1,-2)共线k:1=k+2:-2得到k=-2/3
二,a² = 1 + 1 = 2
b² = 0 + 4 = 4
a · b = 0 - 2 = - 2
|ka-b|=|(k,k+2)|=√[k²+(k+2)²]=√(2k²+4k+4)
|a+b|=|(1,-1)|=√2
θ=120度
(ka - b) · (a + b)
= ka² + ka·b - b·a - b²
= ka² + (k-1) a·b - b²
= 2k - 2(k-1) - 4
= - 2
= |ka-b|·|a+b|·cosθ
整理得 (k+1)²=3
k= -1±√3