If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (–4, 3, –6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.

We need to find the angle between two lines AB and CD but we are not given with the equations this time and we are given with only the points through which these lines start and end.

The coordinates of A, B, C, D are (1,2,3), (4,5,7), (–4,3,–6), (2,9,2).

The direction ratios of line AB are (4–1,5–2,7–3) = (3,3,4)

The direction ratios of line CD are (2+4,9–3,2+6) = (6,6,8)

We can see that the fraction of the corresponding direction ratios will be a constant,

(constant)

Therefore, lines AB and CD are parallel to each other, so the angle between them can be 0⁰ or 180⁰ .