Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

We know that

Two lines with direction ratios a₁, b₁, c₁ and a₂, b₂, c₂ are perpendicular if the angle between them is θ = 90°, i.e. a₁a₂ + b₁b₂ + c₁c₂ = 0

Also, we know that the direction ratios of the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) is taken as x₂ – x₁, y₂ – y₁, z₂ – z₁ (or x₁ – x₂, y₁ – y₂, z₁ – z₂).

⇒ The direction ratios of the line through the points (1, –1, 2) and (3, 4, –2) is:

a₁ = 3 – 1 = 2, b₁ = 4 – (-1) = 4 + 1 = 5, c₁ = -2 –2 = -4

and the direction ratios of the line through the points (0, 3, 2) and (3, 5, 6) is:
a₂ = 3 – 0 = 3, b₂ = 5 – 3 = 2, c₂ = 6 – 2 = 4

Now, consider

a₁a₂ + b₁b₂ + c₁c₂ = 2 × 3 + 5 × 2 + (-4) × 4 = 6 + 10 + (-16) = 16 + (-16) = 0

⇒ The line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).