Lines in space

The essentials

A line in space is fixed by one point on it and its direction. If the line passes through the point with position vector a and is parallel to the vector b, its vector equation is r = a + λb, where λ is a real parameter. Each value of λ gives one point of the line.

If the point is (x₁, y₁, z₁) and the direction ratios of the line are a, b, c, the Cartesian equation is (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. The denominators are the direction ratios of the line.

A line through two points (x₁, y₁, z₁) and (x₂, y₂, z₂) has direction ratios x₂ − x₁, y₂ − y₁, z₂ − z₁, so its equation is (x − x₁)/(x₂ − x₁) = (y − y₁)/(y₂ − y₁) = (z − z₁)/(z₂ − z₁).

Key points

• Direction cosines of a line are the direction cosines of any vector along it (a line has two sets, differing in sign).
• To convert Cartesian form to vector form: a = x₁i + y₁j + z₁k and b = ai + bj + ck.
• A point lies on a line if it gives the same value of λ in every fraction of the Cartesian form.
• Before reading off direction ratios, make sure each numerator is of the form (x − x₁), (y − y₁), (z − z₁) with coefficient +1 for x, y, z.

Exam tips

• Rewrite a term such as (3 − y)/1 as (y − 3)/(−1) before reading the direction ratio.
• To find any point on a line, set each fraction equal to λ and pick a value of λ.

Common mistakes

• Reading direction ratios from a form like (3 − y) without changing the sign.
• Using the coordinates of the point as direction ratios.