Final answer:
Points are collinear if the vectors they form are scalar multiples of each other, indicating that they lie along the same line. Vectors formed between any pairs of points must be parallel, and if they fulfill this criterion through multiplication by a scalar, the points are collinear.
Step-by-step explanation:
To determine whether points are collinear, you can use vectors and analyze their properties. Points are considered collinear if they lie along the same straight line. In vector terms, this means that the vectors formed by any pair of points are scalar multiples of each other (i.e., they have the same or opposite direction), which also implies they are parallel.
When analyzing points using vectors, you can form two vectors using three points, say A, B, and C. If the points are collinear, then the vector ΔAB (vector from A to B) and the vector ΔAC (vector from A to C) would be parallel, and the magnitude of ΔAB times some scalar 'k' should equal the magnitude of ΔAC. If this condition holds, the points are indeed collinear.
To construct such vectors, you can subtract the coordinates of the initial point from the terminal point in each case. If ΔAB equals k times ΔAC, then the direction angles will also be the same or differ by 180°, indicating that the points are collinear.