Question

Check if the line `x+1=1-y=-5z` lies on the plane `2x+3y-5z=1`?

Answer 1

See below.

Explanation:

Given the line with `p=(x,y,z)`

`L->p=p_0+lambda vec v equiv {x+1=1-y=-5z}`

with

`p_0 = (-1,1,0)`
`vec v = (1,-1,-1/5)`

and the plane

`Pi-> << vec n, p-p_1 >> = 2x+3y-5z-1=0`

with

`p_1 = (0,0,-1/5)`
`vec n = (2,3,-5)`

We say that `L in Pi hArr << vec n, p_0+lambda vec v - p_1 >> = 0`

or

`<< vec n, p_0-p_1 >> +lambda << vec n, vec v >> = 0`

Now, substituting values we can conclude.

As can be verified, `<< vec n, p_0-p_1 >> = 0` and `<< vec n, vec v >> = 0` so `L in Pi`