# Vector equations of lines

```Vector equations of lines. 2 – D
Given a point (a,b) and a directional vector < x , y > the vector equation of a line would
be…. _________________________________
Find the vector equation of the line 2x – 4y = 12 use either an x-intercept or y-intercept
as your starting point.
We can also use vector equations to graph other structures.
v (t )  t ,1
or
v (t )  2,t
v (t )  cost ,sint
or
v (t )  t ,t 2
Vector equation of a line in 3 – D.
x0 , y 0 , z 0  t a , b ,c
Parametric form of the line.
x = x0 + at
y = y0 + bt
z = z0 + ct
Symmetric form of a 3 – D line. (Solve for t in each equation and write as a triple
equation).
t 
(x  x 0 )
a
t 
(y  y 0 )
b
t 
(z  z 0 )
c
(x  x0 ) (y  y 0 ) (z  z 0 )


a
b
c
If any of the constants a,b, or c are zero we omit that equation from the equality.
Example:
Given the two 3 – D points (1,-2,3) and (1,2,5), find the following.
a)
b)
c)
d)
Vector Equation
Parametric Equations
Symmetric form of the line.
Does the line pass through the xz plane?
```
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