For each of the following pairs of equations, determine whether they represent the same line, parallel lines, or neither of these.

Question

r=(1,0,3)+s(3,-6,3) and r=(2,-2,5)+t(2,-4,2)

Answer 1

To determine if the two equations represent the same line, we can set them equal to each other and see if there is a solution for s and t:

(1,0,3)+s(3,-6,3) = (2,-2,5)+t(2,-4,2)

We can rewrite this as a system of three equations in the variables s and t:

1 + 3s = 2 + 2t
0 - 6s = -2 - 4t
3 + 3s = 5 + 2t

Simplifying each equation, we get:

3s - 2t = 1
3s + 4t = -2
3s - 2t = 2

We can see that the first and third equations are equivalent, so this system only has two unique equations. To solve for s and t, we can subtract the second equation from either the first or third equation:

-6t = -3
t = 1/2

Plugging this value of t back into one of the equations, we can solve for s:

3s - 2(1/2) = 1
3s - 1 = 1
s = 2/3

So the values of s and t that satisfy both equations are s = 2/3 and t = 1/2. This means that the two equations represent the same line.

We could also see this without solving the system of equations by recognizing that both equations have the same direction vector (3,-6,3), so they must represent parallel lines or the same line. We can then find the specific point on the line by comparing the given points, which are not the same. So we did need to solve the system to confirm that the two equations represent the same line.