Parallel and Orthagonal lines in three dimensions

[ksdhart]

I'm having some trouble understanding this math problem. It says:

51) Find values for \(\displaystyle \alpha\) such that: a) the two lines are parallel; b) the two lines are orthagonal

L₁ = \(\displaystyle <4-t,-6+2t,6+5t>\)
L₂ = \(\displaystyle <\alpha t,2-2\alpha t,3+15t>\)

for any \(\displaystyle t\in \left(\infty ,-\infty \right)\)

The parallel part is easy. I have that taken care of, but I'm struggling with the orthagonal part. As I understand it, two lines are orthagonal if their dot product is zero. So I found the dot product...

L₁ dot L₂ = \(\displaystyle \left(4-t\right)\left(\alpha t\right)+\left(-6+2t\right)\left(2-2\alpha t\right)+\left(6+5t\right)\left(3+15t\right)\)

\(\displaystyle -5\alpha t^2+75t^2+16\alpha t+109t+6=0\)

And that's where I'm having difficulty. It seems like the best I can do it express alpha in terms of t. But since t can take on any real value... I don't know how to proceed. The answer key says the lines are orthagonal when \(\displaystyle \alpha=15\). So that leaves me with:

349t + 6 = 0

Which is only true for one value of t, specifically t = -6/349.

 

[pka]

I'm having some trouble understanding this math problem. It says:
The parallel part is easy. I have that taken care of, but I'm struggling with the orthagonal part. As I understand it, two lines are orthagonal if their dot product is zero. So I found the dot product...

L₁ dot L₂ = \(\displaystyle \color{red}{\left(4-t\right)\left(\alpha t\right)+\left(-6+2t\right)\left(2-2\alpha t\right)+\left(6+5t\right)\left(3+15t\right)}\).

That is completely wrong.

The two direction vectors are \(\displaystyle d_1:<-1,2,5>~\&~d_2=<\alpha,-2\alpha,15>\)
Find \(\displaystyle \alpha\) so that \(\displaystyle d_1\cdot d_2=0\)

 

[ksdhart]

Okay, that does indeed give me the right answer. But why am I using the direction vectors instead of the lines themselves? I think I need more help with this than you can provide. Very very little about this chapter makes any sense. Most of the time, I just do what the book says, or what gets me right answer, but I have no idea why I'm doing it. Thanks for the help though, I do appreciate it.