Let L1 be the line passing through the points Q1=(−1, 5, 3) and Q2=(1, 6, 1) and let L2 be the line passing through the point P1=(3, 10, 2) with direction vector →d=[4, 4, −2]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q. If not, find a value for the z-coordinate of P1 so the resulting lines do intersect.
Did you look at my response to your similar post of this question when you posted as "sam" ??
Follow the same initial steps , with a minor variation at the end.
https://www.jiskha.com/questions/1806307/Let-L1-be-the-line-passing-through-the-points-Q1-4-2-3-and-Q2-0-2-3
To determine whether L1 and L2 intersect, we can set up the following system of equations:
x1 = -1 + t1 * (1 - (-1))
y1 = 5 + t1 * (6 - 5)
z1 = 3 + t1 * (1 - 3)
x2 = 3 + t2 * 4
y2 = 10 + t2 * 4
z2 = 2 + t2 * (-2)
where t1 and t2 are scalar parameters.
Now, we need to find the values of t1 and t2 for which the equations are consistent. This can be done by equating the corresponding components:
-1 + t1 * (1 - (-1)) = 3 + t2 * 4
5 + t1 * (6 - 5) = 10 + t2 * 4
3 + t1 * (1 - 3) = 2 + t2 * (-2)
Simplifying these equations:
2t1 - 4t2 = 4
t1 + 4t2 = 5
-2t1 + 2t2 = -1
Now, we can solve this system of equations. Multiplying the second equation by 2 and adding it to the third equation:
2t1 - 4t2 = 4
2t1 + 8t2 = 10
---------------
12t2 = 14
Solving for t2:
t2 = 14 / 12
t2 = 7 / 6
Substituting this value of t2 into the second equation:
t1 + 4(7/6) = 5
t1 + 14/3 = 5
t1 = 5 - 14/3
t1 = 15/3 - 14/3
t1 = 1/3
Now, we can find the coordinates of point Q by substituting the values of t1 and t2 into the parametric equations of L1:
x1 = -1 + (1/3) * (1 - (-1))
y1 = 5 + (1/3) * (6 - 5)
z1 = 3 + (1/3) * (1 - 3)
x1 = -1 + (1/3) * 2
y1 = 5 + (1/3) * 1
z1 = 3 + (1/3) * (-2)
x1 = -1 + 2/3
y1 = 5 + 1/3
z1 = 3 - 2/3
x1 = 1/3
y1 = 16/3
z1 = 7/3
Therefore, the point of intersection Q is Q = (1/3, 16/3, 7/3).
Since we have found a point of intersection, the lines L1 and L2 do intersect.
Let's move on to finding a value for the z-coordinate of P1 so that the resulting lines intersect. Since we already have the parametric equation for L2, we can substitute the new z-coordinate value and solve for t2:
2 + t2 * (-2) = z
Substituting the value of z = 7/3 (the z-coordinate of Q):
2 + t2 * (-2) = 7/3
Simplifying:
t2 * (-2) = 7/3 - 2
t2 * (-2) = (7 - 6)/3
t2 * (-2) = 1/3
Solving for t2:
t2 = (1/3) / (-2)
t2 = 1/3 * (-1/2)
t2 = -1/6
Therefore, a value for the z-coordinate of P1 that will result in intersection with L1 is z = 7/3.
Please let me know if I can help you with anything else.
To determine whether L1 and L2 intersect, we can first check if they are parallel. If they are not parallel, they will intersect at some point.
Let's start by finding the equations of the lines L1 and L2.
For L1:
We have two points on the line, Q1 and Q2. We can find the direction vector of L1 by subtracting the coordinates of these two points: →d1 = Q2 - Q1.
→d1 = (1, 6, 1) - (-1, 5, 3) = (2, 1, -2).
The equation of L1 can be written as:
L1: (x, y, z) = (x0, y0, z0) + t(2, 1, -2),
where (x0, y0, z0) are the coordinates of any point on the line, and t is a parameter representing any real number.
For L2:
We are given the point P1(3, 10, 2) and the direction vector →d2 = (4, 4, -2).
The equation of L2 can be written as:
L2: (x, y, z) = (3, 10, 2) + s(4, 4, -2),
where (x, y, z) are coordinates of any point on the line, and s is a parameter representing any real number.
Now, let's check if L1 and L2 are parallel. If they are parallel, they will never intersect.
For two lines to be parallel, their direction vectors must be scalar multiples of each other.
So, to check if L1 and L2 are parallel, we need to compare the components of their direction vectors:
The direction vector of L1 is →d1 = (2, 1, -2),
and the direction vector of L2 is →d2 = (4, 4, -2).
Dividing the components of →d2 by 2, we get →d2 = (2, 2, -1).
Comparing the direction vectors →d1 and →d2, we can see that they are not scalar multiples of each other. Therefore, L1 and L2 are not parallel.
Since L1 and L2 are not parallel, they will intersect at some point.
To find the point of intersection Q, we need to find the values of t and s that satisfy the equations of L1 and L2.
Setting the corresponding components equal, we have:
x = 3 + 4s = -1 + 2t,
y = 10 + 4s = 5 + t,
z = 2 - 2s = 3 - 2t.
Solving these equations simultaneously, we can find the values of t and s.
From equation 1: 3 + 4s = -1 + 2t,
From equation 2: 10 + 4s = 5 + t,
Solving these equations, we get s = 2 and t = 0.
Substituting these values back into the equations, we have:
x = 3 + 4(2) = 11,
y = 10 + 4(2) = 18,
z = 2 - 2(2) = -2.
So, the point of intersection Q is (11, 18, -2).
Now, if L1 and L2 were parallel, we would not have a point of intersection. In this case, we need to find a value for the z-coordinate of P1 (3, 10, z) so that the lines intersect.
To determine the value of z, we can set up the following equation:
z = 2 - 2s,
Substituting in the direction vector →d2 = (4, 4, -2), we have:
z = 2 - 2s = 2 - 2(4) = -6.
Therefore, if we set the z-coordinate of P1 to -6, the lines L1 and L2 will intersect.