How can you use vectors to show that points lie on a straight line?

To demonstrate that points lie on a straight line using vectors, the approach involves showing that the vectors representing these points are aligned in the same direction and possess the same gradient. When vectors are in the same direction, it indicates that they can be expressed as scalar multiples of one another. This alignment is crucial because it implies that the points they represent can be connected with a straight line.

Moreover, having the same gradient means that the slope of the line formed by these vectors is consistent throughout the line. Thus, if you were to calculate the gradient between any two points represented by the vectors, it would remain constant, reinforcing the idea that the points lie collinear, or on a single straight path.

In contrast, when vectors do not share the same direction, intersect at right angles, or are merely of equal length without directional alignment, they do not provide sufficient evidence that their corresponding points lie on a straight line. This distinction emphasizes why the correct rationale for showing that points are collinear is rooted in their directional alignment and shared gradient.