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Lines in 3D - Problem 3

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

We’re talking about lines in space. Let’s take a look at anther problem; where does the line <x, y, z> equals <6, 4, 10> plus t times <1, 2, -5> intersect the x,y plane? I’ve drawn a graph here. A quick sketch that shows the <x, y, z> space, the line, the point that it passes through <6, 4, 10>. And I wanted to point out that the x,y plane is down here.

Another way of describing the x,y plane is to call it z equals 0. Of course, the reason for that is that every point on the xy plane has a z coordinate of 0. So that’s an identifying characteristic of the plane. What I’m going to do is I’m going to return this into parametric equations. So I’ll get x equals 6 plus t, y equals 4 plus 2t and z equals 10 plus -5t. I need to find out for what parameter value t, z equals 0, because that’s where the line’s going to cross, the x,y plane.

So I set this equal to 0. I get 10 equals 5t, t equals 2. So I need to plug t equals 2 into these two equations. I get x equals 6 plus 2, 8, and y equals 4 plus, 2 times 2, 4, also 8. And so my coordinates are, <8, 8, 0>. It was really that easy. <8, 8, 0> is the point where this line crosses the x, y plane.

If you wanted to find where it crosses other planes, just use the equation of that plane. Like for example the y, z plane would be equals 0, or the x,z plane would be y equals 0. Just substitute 0 for the appropriate coordinate. Find the parameter value and then find the other coordinates, like we did here.

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