3D Line Strike Plane At point Q 0, the line is parallel to the plane. As time increases, the line intersects theplane at point Q 1, which is on the line because we have a parametric equation forthe 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 oft plus twice y of t plus 4z of t, and that equals minus 1 plus 2t plus twice 2, plus t plus4 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 theconstant 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 whenminus 8t plus 11 equals 7. And that's the same if you manipulate this: You will get tequals 1/2. 𝑄 𝑡 ( )𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 ⇔ − 8𝑡 + 11 = 7 ⇔𝑡 = 12 If you examine the values of 11 and 3, you see that 7 is actually halfway betweenthem. 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 2plus t equals 2 and 1/2 or 5/2. In the second time period, Q is 2 minus 3/2 or 1/2. 𝑄 12 ( ) = 0, 52 , 12 ( ) So we can determine where a line intersects a plane by finding both the parametricequation 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 knowwhere it is.