The line through two points

Exactly one line passes through two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). Its direction ratios can be taken as x₂ − x₁, y₂ − y₁, z₂ − z₁ (or their negatives).

Dividing by the length PQ = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²) gives the direction cosines (x₂ − x₁)/PQ, (y₂ − y₁)/PQ, (z₂ − z₁)/PQ.

Why: drop perpendiculars to the xy-plane and complete a right triangle whose hypotenuse is PQ and whose vertical side is z₂ − z₁; that side over PQ is cos γ. The other two cosines follow the same way.

Worked: through (−2, 4, −5) and (1, 2, 3) the ratios are 3, −2, 8 and PQ = √77, so the direction cosines are 3/√77, −2/√77, 8/√77.

Collinearity: A, B, C lie on one line when AB and BC have proportional direction ratios, because the two segments are then parallel and share B.

Worked: A(2, 3, −4), B(1, −2, 3), C(3, 8, −11) give ratios −1, −5, 7 for AB and 2, 10, −14 for BC; the second is −2 times the first, so the points are collinear.