Need some help with analytic geometry

Question

Check if the following point lies on the plane π : 2x − 3y + 4z − 5 = 0

1. (point) A = (1, -1, 0)

Check if the following vector lies on the plane π : 2x − 3y + 4z − 5 = 0

2. (vector) -AC = (1, 2, 1)

Can someone help me to solve this? With my example please, it would be great.

Answer 1

1. To check whether a point belongs to a plane just substitute x,y,z in you equation by the given numbers. $2*1 - 3(-1) + 4 \cdot0-5=5-5=0$ <- so it does belong to your plane

2. If we want to check whether a vector lies on the plane it has to be perpendicular to the normal vector. To see if it is consider the following vector product: $[2,-3,4] \times [1,2,1]$. If it is zero, then your vector lies on the plane.

Answer 2

Does point A lie in the plane? How would you check that?

Now find a second point in the plane. $A+v.$

Is this point in your plane?

A different way to check to see if your vectors is parallel to your plane. Find the normal vector to your plane. $v$ perpendicular to the normal vector? (How would you check that?)