3D Line Strike Plane At point Q 0, the line is parallel to the plane. As time increases, the line intersects the plane at point Q 1, which is on the line because we have a parametric equation for the line (x = t, y = t2). We can use this equation to determine any point on the line. Let's plug the coordinates of the point into the equation for a plane. So we'll take x of t plus twice y of t plus 4z of t, and that equals minus 1 plus 2t plus twice 2, plus t plus 4 times 2, minus 3t. 𝑥(𝑡) + 2𝑦(𝑡) + 4𝑧(𝑡) = (− 1 + 2𝑡) + 2(2 + 𝑡) + 4(2 − 3𝑡) =− 8𝑡 + 11 If you simplify this a bit, you get 2t plus 2t minus 12t, that would be minus 8t and the constant term is minus 1 plus 4 plus 8 is 11. And we have to compare that with 7. Question is: Is this ever equal to 7? Well, so Q of t is in the plane exactly when minus 8t plus 11 equals 7. And that's the same if you manipulate this: You will get t equals 1/2. 𝑄(𝑡)𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 ⇔ − 8𝑡 + 11 = 7 ⇔𝑡 = 1 2 If you examine the values of 11 and 3, you see that 7 is actually halfway between them. It would make sense that it's halfway in between Q 0 and Q 1, so we will get 7. In the first time period, Q equals minus 1 plus 2t. Plugging in values, we find that 2 plus t equals 2 and 1/2 or 5/2. In the second time period, Q is 2 minus 3/2 or 1/2. ( ) = (0, 𝑄 1 2 5 2 , 1 2 ) So we can determine where a line intersects a plane by finding both the parametric equation of the line and an equation of the plane and substituting one into the other. The time at which this occurs is when the moving point hits the plane, so we know where it is.