Matrices: 2 lines intersect

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October 13th 2008, 11:37 AM
sberxa

Matrices: 2 lines intersect

Okay, so here's the question:
L1:[ 1 2 1 ]^T +s[ 2 -1 1 ] ^T
L2: [ 3 0 1 ]^T +t[ 1 1 2 ]^T
I need to find where they intersect and the cos of the acute angle theta between the lines.
I was thinking of trying to put them into matrices and then trying to equate them, but I don't really know if that's allowed.
I was thinking
L1=[-2 1
1 2
-1 1]
And
L2= [-1 3
-1 0
-2 1]

I was hoping L2-L1 would equal 0, but I have a feeling this is wrong because L1 has the parameter s and L2 has the parameter t.
Any hints are appreciated!
:)
October 13th 2008, 02:26 PM
Jhevon

Quote:

Originally Posted by sberxa

Okay, so here's the question:
L1:[ 1 2 1 ]^T +s[ 2 -1 1 ] ^T
L2: [ 3 0 1 ]^T +t[ 1 1 2 ]^T
I need to find where they intersect and the cos of the acute angle theta between the lines.
I was thinking of trying to put them into matrices and then trying to equate them, but I don't really know if that's allowed.
I was thinking
L1=[-2 1
1 2
-1 1]
And
L2= [-1 3
-1 0
-2 1]

I was hoping L2-L1 would equal 0, but I have a feeling this is wrong because L1 has the parameter s and L2 has the parameter t.
Any hints are appreciated!
:)

you could solve it using matrices, a bit unconventional for this kind of problem, but you can use it. here is the system you want to solve

$L_1:~~~\left< 1 + 2s, 2 - s, 1 + s \right>$
$L_2:~~~\left< 3 + t, t, 1 + 2t \right>$

we want the lines to be equal component-wise in order for them to intersect. thus we need to solve the system

$1 + 2s = 3 + t$ .................(1)
$2 - s = t$ ..........................(2)
$1 + s = 1 + 2t$ .................(3)

so solve that for $s$ and $t$ . you can use matrices if you wish, but as i said. a bit conventional for problems like these...unless this is for a linear algebra class, if it is, use matrices. if it isn't, use another method
October 14th 2008, 12:02 PM
sberxa

Thanks.
So I found the point of intersection to be (11/3,2/3,7/3) using matrices. But how do you find the cosine of theta at these two points? If there was only an x and y value, I guess I could just have made 2 triangles and found the theta of those angles at that point. But how do you do this when there is an (x,y,z) value? Is there a formula to find the distance between two points for a 3D plane? Very confused. Thanks.