Answer:
For three points to be collinear, the vectors connecting the first point to the second point and the first point to the third point must be parallel. That is, the cross product of these two vectors must be equal to the zero vector.
The vector connecting the first point (1, 2, 3) to the second point (4, 7, 1) is:
v = <4-1, 7-2, 1-3> = <3, 5, -2>
The vector connecting the first point (1, 2, 3) to the third point (x, y, 2) is:
w = <x-1, y-2, 2-3> = <x-1, y-2, -1>
To check if these two vectors are parallel, we can take their cross product and see if it is equal to the zero vector:
v x w = <(5)(-1) - (-2)(y-2), (-2)(x-1) - (3)(-1), (3)(y-2) - (5)(x-1)>
= <-5y+12, -2x+5, 3y-5x-6>
For this cross product to be equal to the zero vector, each of its components must be equal to zero. This gives us the system of equations:
-5y + 12 = 0
-2x + 5 = 0
3y - 5x - 6 = 0
Solving this system, we get:
y = 12/5
x = 5/2
Therefore, the values of x and y that make the three points collinear are x = 5/2 and y = 12/5.