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I believe that I have run into a mistake on homework 4. I think that I correctly solved the problem but it was marked wrong. could you guys check me?
Of course, it's hard to check your work without taking a look at it. But we certainly can check to see if this looks correct by simply plotting it. Using math3d.org, I came up with a picture that looks like so:
So, it looks to me like it's just not quite right. It looks like you've got a point on the line but that you don't have the correct direction vector for the line. Double check that, keeping in mind that the direction vector should be the cross product of the two normal vectors for the planes.
Also, when it comes to homework, I absolutely want you to use all resources that you have at your disposal. Certainly, start each problem with pencil and paper and go as far as you can. At some point, though, it's fine to use some computational tools - particularly when it comes to checking answers and computations. To generate the image above, for example, you can use math3d.org like this. To compute a cross product, you could use WolframAlpha.
When I graphed it on math 3D it looked like this
Here is my work and the cross product computed in wolfram alpha.
I'm not sure what I did wrong.
Yes, I missed the 1 in your input - 14 instead of 4.
Looking at the code in the problem, it looks like they might want integers Thus, you could try:
y(t) &= 14t+2 \\
z(t) &= 56t-1. \\