Equation of a Line in Space
A line through point a with direction b has vector equation r = a + t b, and Cartesian form given by equal ratios of the direction cosines.
What is Equation of a Line in Space?
A line through point a with direction b has vector equation r = a + t b, and Cartesian form given by equal ratios of the direction cosines.
Key formula / rule: Direction cosines satisfy l2 + m2 + n2 = 1.
Key points
- Derive the vector and Cartesian equations of a line in space.
- Compute the shortest distance between two skew lines.
Common exam trap
Assuming two non-parallel lines in space must intersect.
Definitions
- Term
Equation of a Line in Space
- Meaning
A line through point a with direction b has vector equation r = a + t b, and Cartesian form given by equal ratios of the direction cosines.
- Term
Equation of a Line in Space — explanation
- Meaning
Direction ratios carry all the directional information, and the angle between two lines is the angle between their direction vectors. Skew lines exist only in three dimensions and need the shortest-distance formula.
Learning objectives
Derive the vector and Cartesian equations of a line in space.
Compute the shortest distance between two skew lines.
Formulae
- Key point
Direction cosines satisfy l2 + m2 + n2 = 1.
- Key point
Two lines are parallel when their direction ratios are proportional.
- Key point
Skew lines are neither parallel nor intersecting.
Prerequisites
MAT-U4-VA-T1-S1-C3
Common mistakes
Assuming two non-parallel lines in space must intersect.
Using direction ratios as direction cosines without normalising.
Keywords
Equation
Line
Space
Practice preview
Find the image of the point P(1, 6, 3) in the line x/1 = (y - 1)/2 = (z - 2)/3.…
hard
What is the vector equation of a line passing through the point (1, 2, 3) and parallel to the vector 3i + 2j - 2k?…
easy
Does the point (2, -1, 3) lie on the line given by the equation (x - 1)/1 = (y + 2)/(-1) = (z - 4)/1?…
easy