Parametric Representation of a Straight Line:

For the position vectors P₁ [1 2] and P₂ [4 3], find out the parametric representation of the line segment among them. Also find out the slope and tangent vector of the line segment.

Solution

A parametric representation is following

P (t) = P₁ + (P₂ - P₁) t = [1 2] + ([4 3] - [1 2]) t            0 ≤ t ≤ 1

P (t) = [1 2] + [3 1] t    0 ≤ t ≤ 1

Parametric representations of x & y components are following

x (t) = x₁ + (x₂ - x₁) t = 1 + 3t   0 ≤ t ≤ 1

y (t) = y₁ + (y₂ - y₁) = 2 + t

The tangent vector is got by differentiating P (t). Specifically,

P' (t) = [x' (t)  y' (t)] = [3  1]

or     = 3i +  j

where,is the tangent vector and i, j are unit vectors respectively in the x, y directions.

The slope of the line segment is

dy/dx = (dy /dt )/ (dx/ dt)   = y′(t) / x′(t)  = 1/3