Question

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

Answer 1

Given points are:

⇒ A = (1, 2, 3)

⇒ B = (4, 5, 7)

⇒ C = (–4, 3, –6)

⇒ D = (2, 9, 2)

Let us assume the direction ratios for line AB be (r₁, r₂, r₃) and CD be (r₄, r₅, r₆)

We know that direction ratios for a line passing through points (x₁, y₁, z₁) and (x₂, y₂, z₂) is (x₂ – x₁, y₂ – y₁, z₂ – z₁).

Let’s find the direction ratios for the line AB

⇒ (r₁, r₂, r₃) = (4 – 1, 5 – 2, 7 – 3)

⇒ (r₁, r₂, r₃) = (3, 3, 4)

Let’s find the direction ratios for the line CD

⇒ (r₄, r₅, r₆) = (2 – (– 4), 9 – 3, 2 – (– 6))

⇒ (r₄, r₅, r₆) = (2 + 4, 9 – 3, 2 + 6)

⇒ (r₄, r₅, r₆) = (6, 6, 8)

We know that the angle between the vectors with direction ratios proportional to (a₁, b₁, c₁) and (a₂, b₂, c₂) is given by:

Using the above formula we calculate the angle between the vectors.

Let α be the angle between the two vectors given in the problem.

∴ The angle between the given two vectors is \(0^\circ\).