1. Determine if the following equations represent the same line
L1=(0,-2,3) + t(-1,1,-2)
L2=(-1,1,1)+ v(-2,2,-4)
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To determine if the two equations represent the same line, we need to check if they have the same direction vector and a common point.
Let's start by finding the direction vectors of each line. For L1, the direction vector is (-1, 1, -2), and for L2, the direction vector is (-2, 2, -4).
The direction vectors for L1 and L2 are scalar multiples of each other if they have the same ratios for each component. We can check this by dividing the components of one vector by the corresponding components of the other vector and checking if the ratios are the same.
For L1 and L2, the ratios for the x, y, and z components are (-1)/(-2) = 1/2, 1/1 = 1, and (-2)/(-4) = 1/2, respectively.
Since the ratios are the same for all three components, we can conclude that the direction vectors are scalar multiples of each other.
Now, we need to check if the lines have a common point. We can do this by equating the equations of the lines and solving for t and v.
For L1, the coordinates of an arbitrary point on the line are given by (0, -2, 3) + t(-1, 1, -2).
For L2, the coordinates of an arbitrary point on the line are given by (-1, 1, 1) + v(-2, 2, -4).
Equating the coordinates, we have:
0 - t = -1 - 2v (for the x-coordinate)
-2 + t = 1 + 2v (for the y-coordinate)
3 - 2t = 1 - 4v (for the z-coordinate)
Simplifying these equations, we get:
-t + 2v = -1 (equation 1)
t - 2v = 3 (equation 2)
2t + 4v = 2 (equation 3)
We can solve this system of equations to find the values of t and v.
By adding equations 1 and 2, we eliminate v:
2t = 2
So, t = 1.
Substituting t = 1 into equations 1 and 2, we can solve for v:
-1 + 2v = -1
2v = 0
v = 0
Therefore, t = 1 and v = 0 satisfy the system of equations.
Since the lines have the same direction vector and a common point, we can conclude that the equations represent the same line.