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].
a) Show that the lines intersect at the right angle.
b) find the point of intersection.
please and thanks!
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.
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
Yes, all of the above. But you need to check that the components (z) match "automatically".
You're welcome.
how do i check that? that the components (z) match "automatically"
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.
a) To determine if the lines intersect at a right angle, we can use the dot product of the direction vectors of the lines. If the dot product is 0, then the lines are perpendicular.
Let's find the direction vectors of the lines:
Line 1: [x, y, z] = [4, 7, -1] + t[4, 8, -4]
The direction vector of this line is [4, 8, -4].
Line 2: [x, y, z] = [1, 5, 4] + s[-1, 2, 3]
The direction vector of this line is [-1, 2, 3].
Now, let's calculate the dot product of the direction vectors:
Dot product = (4 * -1) + (8 * 2) + (-4 * 3)
= -4 + 16 - 12
= 0
Since the dot product is 0, we can conclude that the lines intersect at a right angle.
b) To find the point of intersection, we need to solve the system of equations formed by equating the two parametric equations of the lines.
Let's equate the x, y, and z components:
For x: 4 + 4t = 1 - s
For y: 7 + 8t = 5 + 2s
For z: -1 - 4t = 4 + 3s
We can solve this system of equations to find the values of t and s. Substituting the value of s from the first equation into the other two equations:
7 + 8t = 5 + 2(4 + 3s) (equation 2)
-1 - 4t = 4 + 3(4 + 3s) (equation 3)
Simplifying equation (2):
7 + 8t = 5 + 8 + 6s
8t - 6s = 0 (equation 4)
Simplifying equation (3):
-1 - 4t = 4 + 12 + 9s
-4t - 9s = 17 (equation 5)
Now we have two equations (4) and (5) to solve the system.
Using any suitable method (such as substitution or elimination), we can solve equations (4) and (5) to find the values of t and s.
Once we have the values of t and s, we substitute them back into one of the given lines' equations to find the point of intersection.
I hope this helps you find the point of intersection!