Determine whether the lines intersect(have a common point) and if so, find the coordinates of that point?
r(t) = < 4t+3, 4t + 2, 6t + 6>, for -infinity < t < infinity
R(s) = < s + 2, 3s - 1, -4s + 10>, for -infinity < t < infinity
And if r and R describe the paths of two particles, do the particles collide?
My teacher did an example in class but this question differs from this one so I don't know how to do it.
let's re-write each line as
x = 4t+3
y = 4t+2
z = 6t+6
and
x = s+2
y = 3s-1
z = -4s + 10
so
4t+3 = s+2 ---> 4t - s = -1
4t+2 = 3s-1 --> 4t - 3s= -3
subtract: 2s= 2
s = 1 , then t=0
if 2nd line, when s = 1
x = 3
y=2
z = 6
in 1st line when t=0
x=3
y=2
z=6
Yes, they do intersect at (3,2,6)
To determine if the lines intersect, we need to find a common point that satisfies both equations. Let's set up the equations for the x, y, and z coordinates:
Equation for r(t):
x = 4t + 3
y = 4t + 2
z = 6t + 6
Equation for R(s):
x = s + 2
y = 3s - 1
z = -4s + 10
We can equate the x, y, and z values from the two equations to find potential values of t and s that satisfy both lines.
1) Equating the x-coordinates:
4t + 3 = s + 2
2) Equating the y-coordinates:
4t + 2 = 3s - 1
3) Equating the z-coordinates:
6t + 6 = -4s + 10
Now, we can solve these equations simultaneously to find the values of t and s:
1) From equation 1:
4t = s - 1
t = (s - 1) / 4
2) Substituting t into equation 2:
4(s - 1) / 4 + 2 = 3s - 1
s - 1 + 2 = 3s - 1
3 = 2s
s = 3 / 2
3) Substituting s back into equation 1:
t = ((3 / 2) - 1) / 4
t = -1 / 8
Now we have values for t and s. Substituting them into any of the original equations will give us the coordinates of the common point where the lines intersect.
Using equation r(t):
x = 4(-1/8) + 3 = 2.5
y = 4(-1/8) + 2 = 1.5
z = 6(-1/8) + 6 = 5.25
Therefore, the lines intersect at the point (2.5, 1.5, 5.25).
To determine if the particles collide, we need to consider their paths. Given that r(t) and R(s) describe the paths of two particles, if the two particles intersect at the same point (2.5, 1.5, 5.25) at any given time, then they collide.
To determine whether the lines intersect and find the coordinates of that point, we need to find the values of t and s for which the coordinates of the two lines are equal.
Comparing the x-component of the two lines gives us:
4t + 3 = s + 2
Comparing the y-component of the two lines gives us:
4t + 2 = 3s - 1
Comparing the z-component of the two lines gives us:
6t + 6 = -4s + 10
Now we have a system of three equations with two unknowns, t and s. To solve for t and s, we can solve this system of equations using any method, such as substitution or elimination. Let's use substitution:
From the first equation, we can express s in terms of t:
s = 4t + 3 - 2
s = 4t + 1
Substituting this expression for s into the second equation:
4t + 2 = 3(4t + 1) - 1
4t + 2 = 12t + 3 - 1
4t + 2 = 12t + 2
-10t = 0
t = 0
Plugging t = 0 into the expression for s:
s = 4(0) + 1
s = 1
Therefore, the lines intersect at the point where t = 0 and s = 1.
To find the coordinates of the point of intersection, we substitute these values of t and s into any of the initial parametric equations. Let's use r(t):
r(0) = <4(0) + 3, 4(0) + 2, 6(0) + 6>
r(0) = <3, 2, 6>
So, the coordinates of the point of intersection are (3, 2, 6).
Now, to determine if the two particles collide, we need to check if their paths cross at any point. Since the point of intersection we found lies on both paths, the two particles will collide at the coordinates (3, 2, 6).