Can you please check whether my answers are correct? If they aren't, how would you find if the points are parallel to one another with 3 coordinates?

Using vectors, demonstrate that these points are collinear.

a)
P(15 , 10)
Q(6 , 4)
R(-12 , -8)

Vector PQ = (-9 , -6)
Vector QR = (-18 , -12)
Vector RP = (27 , -18)

(-9 / 27 / -18) = (-6 / -18 / -12)

Therefore, not collinear.

b)
D(33, -5, 20)
E(6, 4, -16)
F(9, 3, -12)

Vector DE = (-27, 9, -36)
Vector EF = (3, -1, 4)
Vector FD = (24, -8, 7)

(-27 / 3 / 24) = (9 / - 1 / -8) = (-36 / 4 / 7)

Therefore, not collinear.

no

Vector PQ = (-9 , -6)
Vector QR = (-18 , -12)

so vector QR = 2(-9,-6) = 2PQ

so P,Q and R are collinear since we were able to express them as a linear combination.

same thing in b)
notice that (-27,9,-36) = -9(3,-1,4)

To determine if the given points are parallel to each other with their coordinates, you can follow these steps:

1. Calculate the vectors between each pair of points by subtracting the coordinates of the starting point from the coordinates of the ending point.

2. Simplify the vectors to their simplest form by dividing each component by their greatest common divisor (GCD).

3. Check if the ratios of the components of the vectors between each pair of points are equal.

Let's apply this method to the given points:

a)

P(15, 10)
Q(6, 4)
R(-12, -8)

Vector PQ = Q - P = (6 - 15, 4 - 10) = (-9, -6)
Vector QR = R - Q = (-12 - 6, -8 - 4) = (-18, -12)
Vector RP = P - R = (15 - (-12), 10 - (-8)) = (27, 18)

To simplify the vectors:
Vector PQ = (-9 / 3, -6 / 3) = (-3, -2)
Vector QR = (-18 / 6, -12 / 6) = (-3, -2)
Vector RP = (27 / 9, 18 / 9) = (3, 2)

The ratios of the components of all the vectors are equal: -3/-3 = -2/-2 = 3/3 = 2/2 = 1.

Therefore, the points P, Q, and R are collinear and parallel to each other.

b)

D(33, -5, 20)
E(6, 4, -16)
F(9, 3, -12)

Vector DE = E - D = (6 - 33, 4 - (-5), -16 - 20) = (-27, 9, -36)
Vector EF = F - E = (9 - 6, 3 - 4, -12 - (-16)) = (3, -1, 4)
Vector FD = D - F = (33 - 9, -5 - 3, 20 - (-12)) = (24, -8, 32)

To simplify the vectors:
Vector DE = (-27 / -3, 9 / 3, -36 / -3) = (9, -3, 12)
Vector EF = (3 / 1, -1 / -1, 4 / 1) = (3, 1, 4)
Vector FD = (24 / 8, -8 / -1, 32 / 8) = (3, 8, 4)

The ratios of the components of the vectors are not equal: 9/3 ≠ -3/1 ≠ 12/4.

Therefore, the points D, E, and F are not collinear and not parallel to each other.

In summary:
a) Points P, Q, and R are collinear and parallel to each other.
b) Points D, E, and F are not collinear and not parallel to each other.