# Lines in a Plane 13

• Sep 6th 2009, 08:28 AM
skeske1234
Lines in a Plane 13
Find a vector equation for the line through the origin that intersects both of the lines r=(2,-16,19)+t(1,1,-4) and r=(14,19,-2)+u(-2,1,2).

This is my work so far, but I can't seem to find the correct answer.

d1=(1,1,-4)
d2=(-2,1,2)

i'm stuck here..
• Sep 6th 2009, 11:31 AM
running-gag
Let r=(0,0,0)+v(a,b,c) the requested line

There exists a value for v, say $\displaystyle v_1$ and another value, say $\displaystyle v_2$ such that

$\displaystyle 2+t=v_1a$
$\displaystyle -16+t=v_1b$
$\displaystyle 19-4t=v_1c$
$\displaystyle 14-2u=v_2a$
$\displaystyle 19+u=v_2b$
$\displaystyle -2+2u=v_2c$

Let $\displaystyle x = \frac{v_2}{v_1}$

$\displaystyle 14-2u=x(2+t)$
$\displaystyle 19+u=x(-16+t)$
$\displaystyle -2+2u=x(19-4t)$

Solving gives
$\displaystyle x = -\frac{64}{9}$
$\displaystyle u = 41$
$\displaystyle t = \frac{121}{16}$

You can chose one of the coordinates a, b or c
If you chose a = 17 then b = -15 and c = -20
• Sep 6th 2009, 02:07 PM
skeske1234
how did you get a=17?
• Sep 7th 2009, 06:34 AM
skeske1234
I can't seem to figure out how you got your final answer, the a=17 then b=-15 and c =-20.. this step before this, or to get to this a=17, where did it come from?

Can you, running-gag, or someone else clarify it for me? please and thank you
• Sep 7th 2009, 10:41 AM
running-gag
As I said before you can chose any value for a
I have chosen a=17 because it makes the other values very simple

The coordinates of the intersect with r=(2,-16,19)+t(1,1,-4) is given for t=121/16, therefore (153/16,-135/16,-180/16)

If you chose a=17, you will get b=-15, c=-20 and v1=9/16
But if you chose a=34, you will get b=-30, c=-40 and v1=9/32

This does not change the coordinates of the intersect (v1a,v1b,v1c) and does not change either the line since r=(0,0,0)+v(17,-15,-20) and r=(0,0,0)+v(34,-30,-40) are the same line
• Sep 7th 2009, 01:20 PM
skeske1234
ok, so i could even choose a=1 and then find v1, and then find b and c right?
so
a=1
v1=153/16

b=-135/153

c=-180/153
• Sep 8th 2009, 09:15 AM
skeske1234
running-gag? can you let me know about what i said above? just this last question

Thanks so much for your help by the way :)
• Sep 8th 2009, 10:47 AM
running-gag
That's correct ! (Clapping)