Answer: To determine whether the lines x=3+2t, y=2-2t, z=-1+3t are the same line, parallel lines, or intersecting lines, we need to compare their direction vectors.
The direction vector of a line is the coefficients of t in the parametric equations.
For the given lines:
- The direction vector of the first line is (2, -2, 3).
- The direction vector of the second line is also (2, -2, 3).
Since the direction vectors of the two lines are the same, the lines are either the same line or parallel lines.
To determine whether they are the same line or parallel lines, we can compare a point on each line.
For the first line, let's consider t = 0:
- x = 3 + 2(0) = 3
- y = 2 - 2(0) = 2
- z = -1 + 3(0) = -1
So, a point on the first line is (3, 2, -1).
For the second line, let's consider t = 0:
- x = 3 + 2(0) = 3
- y = 2 - 2(0) = 2
- z = -1 + 3(0) = -1
So, a point on the second line is also (3, 2, -1).
Since both lines have the same direction vector and pass through the same point, the lines are the same line.
Explanation: