10. Consider the lines [x,y,z] = [4,7,-1] + t[4,8,-4] and [x,y,z] = [1,5,4] + s[-1,2,3].

Question

a) Show that the lines intersect at the right angle.

b) find the point of intersection.

please and thanks!

Answer 1

First see that the two vectors are orthogonal:
[4,8,-4].[-1,2,3]=-4+16-12=0 => orthogonal

Next solve for s and t for each coordinate direction (x,y,z)
4+4t=1-s
7+8t=5+2s
=> s=-1, t=-1/2
Substitute into original equation:
<4,7,-1>-<2,4,-2>=<2,3,1>
<1,5,4>-<-1,2,3>=<2,3,1>
<2,3,1> is the intersection of the two lines.
If the two lines do not cross (but are still orthogonal), the third component (z) will not match.

Answer 2

is this

a) First see that the two vectors are orthogonal:
[4,8,-4].[-1,2,3]=-4+16-12=0 => orthogonal

Next solve for s and t for each coordinate direction (x,y,z)
4+4t=1-s
7+8t=5+2s
=> s=-1, t=-1/2

and this

b) Substitute into original equation:
<4,7,-1>-<2,4,-2>=<2,3,1>
<1,5,4>-<-1,2,3>=<2,3,1>
<2,3,1> is the intersection of the two lines.
If the two lines do not cross (but are still orthogonal), the third component (z) will not match

? thank you

Answer 3

Yes, all of the above. But you need to check that the components (z) match "automatically".

You're welcome.

Answer 4

how do i check that? that the components (z) match "automatically"

Answer 5

From
"the lines [x,y,z] = [4,7,-1] + t[4,8,-4] and [x,y,z] = [1,5,4] + s[-1,2,3]. "
We solved for s and t using the x- and y-coordinates. We need to check the z-coordinates using:
s=-1, t=-1/2
First equation: -1+(-4t) = -1+(-1/2)*-4=1
Second equation: 4+3s = 4+3(-1) = 1

So they match (for the two given equations) which means that the two lines actually meet, not one on top of the other.