Does the plane intersects the points?

Question

I have a plane, which intersects the point P(1,2,0) and his normal vector is n=(5,-2,-1). Does the plane intersects the points A(3,-2,1) and B(-2,5,2). If it intersects then calculate the intersection's coordinates.

How to calculate it? Thank You for your time and contribution!

Answer 1

I think here you mean that (3,-2,1)and (-2,5,2) lie on the plane or not...I can be solved easily by considering a point on plane Q(x,y,z) and as already a point on the plane is given as P(1,2,0) the PQ vector becomes (x-1, y-2, z-0) and since it would always be perpendicular to the normal vector of the plane hence the dot product of PQ.n = 0 as cos(90 degrees)=0 and from here simple the equation of the plane pops out as 5x-2y-z-z = 0. Then for the respective point put the values of x, y, z and if they satisfy the equation, they lien on the plane or otherwise they don't lie...

Answer 2

The planes that are perpendicular to a normal vector $(a,b,c)$ have equation

$ax + by + cz = k$

for some constant $k$. If you are given the co-ordinates $(x_0, y_0, z_0)$ of a point on the plane then you can substitute $x_0$, $y_0$ and $z_0$ into this equation to find the value of $k$.

Then you can substitute the co-ordinates of other points into this equation (keeping the same value of $k$ that you have just determined) to see whether or not they lie on this plane.