已知向量a=(x,4,1)向量b=(-2,y,-1)向量c=(3,-2,z)向量a∥向量b向量c⊥向量b求1.向量bac的坐标2（向量a＋向量c﹚与﹙向量b＋向量c﹚的夹角的余弦值是多少

Question

已知向量a=(x,4,1) 向量b=(-2,y,-1)向量c=(3,-2,z) 向量a∥向量b 向量c⊥向量b
求1.向量b a c 的坐标 2 （向量a＋向量c﹚与﹙向量b＋向量c ﹚的夹角的余弦值是多少

Answer 1

∵ a//b 且 a = (x,4,1),b = (-2,y,-1)
∴ a = -b
∴ x = 2,y = -4
∵ c⊥b
∴ c·b = -6 - 2y - z = 0
∴ z = -6 - 2y = -6 + 8 = 2
∴ a (2,4,1),b (-2,-4,-1),c (3,-2,2)
a + c = (2,4,1) + (3,-2,2) = (5,2,3)
b + c = (-2,-4,-1) + (3,-2,2) = (1,-6,1)
∴ (a+c)·(b+c) = (5,2,3)·(1,-6,1) = 5 - 12 + 3 = -4
∴ |a+c| = √(25 + 4 + 9) = √38
∴ |b+c| = √(1 + 36 + 1) = √38
∴ 根据公式 x·y = |x| |y| cosA
∴ cosA = -4 / (√38)² = -4 / 38 = -2/19