Question

一个数学直线方程的问题L1:A1x+B1y+C1=0 和L2：A2x+B2y+C2=0中A1A2乘B1B2=0求证L1垂

Answer 1

A1A2 + B1B2 = 0,

若A1 = 0,

则B1B2 = 0,

若B1 = 0,

则直线L1失去意义,所以,A1 = 0时,B1一定不等于0,所以B2 = 0.

同样,B2 = 0时,A2一定不等于0.

这时,

L1:B1y + C1 = 0,平行于X轴

L2:A2x + C2 = 0,平行于Y轴

L1,L2相互垂直.

同理,

若A2 = 0,

则,B2不等于0,B1 = 0,A1不等于0.

这时,

L1:A1x + C1 = 0,平行于Y轴

L2:B2y + C2 = 0,平行于X轴

L1,L2相互垂直.

当A1,A2都不等于0时,B1,B2也都不等于0.

A1A2 + B1B2 = 0,

(A1/B1)(A2/B2) + 1 = 0,

(A1/B1)(A2/B2) = -1.

此时,

L1的斜率 = -A1/B1,

L2的斜率 = -A2/B2.

L1的斜率*L2的斜率 = (A1/B1)(A2/B2) = -1.

L1与L2相互垂直.

综合,知,

当A1A2 ＋ B1B2 ＝ 0时,总有L1与L2相互垂直.