How To Determine If Vector Intersects A Plane Defined as 'z = 0'

Question

So I'm rather new to vector/3d mathematics so i'm probably missing something but...

If we have a point p at location (0, 0, 0) (the origin) from which a vector v departs, which is defined as <0, 0, 1> (so the vector only travels in the z-direction).

To get the plane of which v is the normal-vector I could use the following formula:

• a1(x - x0) + a2(y - y0) + a3(z - z0) = 0

if we fill it in we get:

• 0(x - 0) + 0(y - 0) + 1(z - 0) = 0
• (z-0) = 0
• z = 0

This plane should be a plane that spans the entirity of the x and y axis and does not move into the z-direction at all. Now, if we have the point p2 = (1, 2, 3) with vector v2 = <0, 0, 1> then we know that v2 from p2 should not intersect the plane that we have made. Yet, when I parametrize v2, fill the z in in the plane-formula then I get a formula that is solvable;

V2 parametrized is:

• <1 + 0t, 2 + 0t, 3 + 1t>

If we substitute x, y and z in our plane-formula then we get...

• 3 + 1t = 0
• t = -3

And if I feed this back into the parametrized vector then the point of intersection is:

• (1 + 0*-3, 2 + 0*-3, 3 + 1*-3) =
• (1, 2, 0)

The problem here is, there should be no point of intersection at all... What am I doing wrong?

Sorry if I made any noob-errors.

Answer 1

Your calculations are right, it does indeed intersect. Try drawing a picture to help your intuition! In this case, the vector emanating from $p_2$ moves up and down only in the z-direction, so must intersect the plane $z=0$ at some point

This is guaranteed because $v2$ is the normal vector, if you want to guarantee that the line doesn't intersect the plane, try using a parallel vector

Answer 2

You’ve computed the intersection of the line through $p_2$ with direction $v_2$ with the plane. Note that the value of $t$ that you got is negative, which means that the intersection point is in the opposite direction of $v_2$ from $p_2$. For the ray defined by this point and vector to intersect a plane, you must have $t>0$.