Using vectors show that the points `A(-2,3,5),B(7,0,-1)C(-3,-2,-5)` and `D(3,4,7)` are such that `A B` and `C D` intersect at the point `P(1,2,3)dot`

Explanation

Here, `vec(AP) = vecp-vecA = 3hati-hatj-2hatk`

`vec(AB) = 9hati-3hatj-6hatk = 3(3hati-hatj-2hatk) = 3vec(AP)`

So, `A,P and B` are collinear.

Now, `vec(CP) = 4hati+4hatj+8hatk`

`vec(CD) = 6hati+6hatj+12hatk = 3/2(4hati+4hatj+2hatk) = 3/2vec(CP)`

So, `C,P and D` are collinear.

It means `AB` and `CD` intersects at point `P`.