Question

How to prove by vectors that two lines are intersecting?

Answer 1

If the perpendicular distance between 2 lines is zero, then they are intersecting.

Let r1=a1+xb1r1=a1+xb1

And r2=a2+yb2r2=a2+yb2

Here r1 and r2 represent the 2 lines , and a1, a2, b1, b2 are vectors. x and y are constants. These two equations represent the vector equation of the given lines.

The shortest distance between two skew lines is given by

d=|(a1−a2).(b1d=|(a1−a2).(b1 X b2)|/|b1b2)|/|b1 Xb2b2|

Where d is the shortest distance between them. Now d = 0, as they are intersecting,

So, |(a1−a2).(b1|(a1−a2).(b1 X b2b2)|=0)|=0

Thus, to prove two lines are intersecting, plug in the values in the above given expression and show that it is equal to zero.