3D Line Strike Plane
At point Q 0, the line is parallel to the plane. As time increases, the line intersects thenplane at point Q1, which is on the line because we have a parametric equation fornthe 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 ofnt plus twice y of t plus 4z of t, and that equals minus 1 plus 2t plus twice 2, plus t plusn4 times 2, minus 3t. x(t) + 2yt ( ) + 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 thenconstant 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 whennminus 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 betweennthem. 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. Q(1/2)=O(5/2, 1/2) So we can determine where a line intersects a plane by finding both the parametricnequation of the line and an equation of the plane and substituting one into the other.nThe time at which this occurs is when the moving point hits the plane, so we knownwhere it is.
