Question

What are the parametric equations for the line of intersection of the planes `x + y + z = 7` and `x + 5y + 5z = 7`?

Answer 1

`x=7`, `y=t` and `z=-t`

Explanation:

The two planes have normal vectors `vecu<1,1,1>` and `vecv<1,5,5>`.

Let `l` be the line of intersection. Then we will have

`l`||`(vecuxxvecv)`

`(vecuxxvecv)=<(1xx5-1xx5),-(1xx5-1xx1),(1xx5-1xx1)>=<0,-4,4>`

Now let us find a point `P` on line `l`, say where it cuts plane `z=0`

then `x+y=7` and `x+5y=7=>` `x=7` and `y=0`

hence `P` is `(7,0,0)` and equation of line is

`x=7` and `y+z=0`

and parametric equation of line is

`x=7`, `y=t` and `z=-t`