Parametric equation of a line
r = a + t d — a point on the line plus any multiple of a direction vector.
Key formulas
Reading the parts
a is the position vector of a known point. d points along the line. Parameter t runs through all real numbers.
Example
Through (1,2) with direction (3,−1): (x,y)=(1,2)+t(3,−1).
Convert to Cartesian
Eliminate t: from x=1+3t and y=2−t get (x−1)/3 = 2−y.
Worked example
Worked example
Problem. A line passes through (1, 2) with direction vector (3, −1). Write a parametric equation and eliminate t.
Solution. (x, y) = (1, 2) + t(3, −1). So x = 1 + 3t, y = 2 − t → t = 2 − y = (x−1)/3 → (x−1)/3 = 2 − y.
Parallel
Same d (or a scalar multiple) means parallel lines.
Review quiz
Through (0,−4) with direction (5,2). When t=2 the point is:
Practice questions
- Write a parametric form through (0,−4) with direction (1,2).
- Eliminate t from x=2+t, y=−1+4t.
Study notes
Is t unique for a point?
Yes for a fixed choice of a and d.