Parametric representation of lines

Page 1

To recap instead of writing vectors as a = (3, 4) for simplification unit vectors can be used as i = (1, 0) and j = (0, 1). As unit vectors, a would equal 3*i + 4*j or 3i + 4j

Example:

v = (2, 1)

2v

1.5v

v & 2v & 1.5v & -1v are all on the same line/are all co-linear

S= {c of v | c as element of R} ---> Set of co-linear vectors

x = (2, 4) representation in standard form

L = {x + t*v | t element of R} -----> allows extension to infinite number of dimensions

Example 2:

a = (2, 1) & b = (0, 3)

Equation for line that cuts through both vector ends ---> Don't forget to shift by adding vector a or b

a/b + t(b - a) =(-2, 2) > allows multiplication by any scalar e.g. 2, 3, 400, 3021, -21, -231432, -31243523

In order to get a vector that lies on two factors you need to shift by e.g.

a or b, hence -----> Set = {a/b +t*(a+b/a-b) | t is an element of R}

L = (0,3) + (-2, 2) = -2t + 5t

Example 3 in R3:

P1 = (-1, 2, 7) & P2 = (0,3,4) in the equation L = {P1 or P2 + t*(P1-P2)|t element of R}

L = (-1, 2, 7) + t(-1,2,7)-(0,3,4) = (-1, 2, 7) + (-1, -1, 3) = (-2, 1, 10)

Has to be written as parametric equation to define a line

x = -1 + -1t

y = 2 -1t

z + 7+3t