Question

已知向量A（3，2),B(-1,2)C(4,1)设向量D（x,y)满足（向量D-C)//(向量A+B)且向量D-C的绝对值=1求向量D

Answer 1

D-C = (x-4, y-1)
A+B = (2,4)
(y-1) / (x-4) = 4/2 = 2
y = 2x -7
(x-4)^2 + (y-1)^2 = 1
x = 4+ 根号5 / 5 或 4- 根号5 / 5
y = 1+ 2根号5 / 5 或 1- 2根号5 / 5
D = (4+ 根号5 / 5 ,1+ 2根号5 / 5) 或 (4- 根号5 / 5 ,1- 2根号5 / 5)

Answer 2

(1) a = mb + nc
(3, 2) = m(-1, 2) + n(4,1)
(3, 2) = (-m + 4n, 2m + n)
∴ -m + 4n = 3, 2m + n = 2
联立方程得 m = 5/9, n = 8/9

(2) a + kc = (3, 2) + k(4, 1)
= (3 + 4k, 2 + k)
2b - a = 2(-1, 2) - (3, 2)
= (-5, 2)
∵ (a + kc) // (2b - a)
∴ 2(3 + 4k) - (-5)(2 + k) = 0
解得 k = -16/13

(3) d - c = (x, y) - (4, 1)
= (x - 4, y - 1)
a + b = (3, 2) + (-1, 2)
= (2, 4)
∵ (d - c) // (a + b)
∴ 4(x - 4) - 2(y - 1) = 0
2x - y - 7 = 0 ~ (1)

且∣d - c∣ = 1
∴√ [(x - 4)² + (y - 1)² ] = 1
两边平方, (x - 4)² + (y - 1)² = 1 ~ (2)

联立(1), (2)式, 得 x = 4±√ 5/5 y = 1±2√5/5

解得 d = (4+√ 5/5, 1+2√ 5/5) 或 (4-√ 5/5, 1-2√ 5/5)