(Solved): Determine whether the following lines in R2 represented by the vector equations below intersect, ar ...

Expert Answer

---

There are two questions each containing 3 parts now first we solve 1 st question:-

Apologies for the errors in the previous response. Let's reanalyze the given vector equations.
  
The direction vector of line r is ?-1, 2?, and the direction vector of line s is ?-3, 1?. The direction vectors are not scalar multiples of each other, so the lines are not parallel.
To check if they intersect, we set the x-components and y-components equal to each other and solve for t:

-2 - t = -2 - 3t => 2t = 0 => t = 0
5 + 2t = 5 + t => 2t = t => t = 0

The values of t are the same for both equations, indicating that the lines intersect at the point (-2, 5) when t = 0. Therefore, the lines intersect.

2 .  

The direction vector of line r is ?-3, 2?, and the direction vector of line s is ?6, -4?. The direction vectors are not scalar multiples of each other, so the lines are not parallel.

To check if they intersect, we set the x-components and y-components equal to each other and solve for t:

2 - 3t = 2 + 6t => -9t = 0 => t = 0
5 + 2t = 6 - 4t => 6t = 1 => t = 1/6

The values of t are different for both equations, indicating that the lines do not intersect. Therefore, the lines are parallel.