VECTOR and PARAMETRIC EQUATIONS
(LINES in R2)
A.
DIRECTION VECTORS
To determine the equation of a straight line,
the following is required:
a)
b)
two points on the line OR
one point on the line and its direction.
Any vector parallel to a line may be used as
a direction vector, 𝒎.
Ex 
B.
State a direction vector for each of the following:
a)
the line through A(1,6) and B(4,0);
b)
the line with slope ;
_______________
c)
a horizontal line;
_______________
d)
a vertical line.
_______________
4
5
_______________
VECTOR EQUATION of a LINE (R2)
Given direction vector, 𝑚 = (𝑎, 𝑏), and point, 𝑃𝑜 (𝑥𝑜 , 𝑦𝑜 ), a line is uniquely determined
through 𝑃𝑜 and parallel to 𝑚:
𝑚
P(x,y)
𝑃𝑜 𝑃 = 𝑡𝑚
b
a
Po(xo,yo)
VECTOR EQUATION of a LINE in R2:
𝒙, 𝒚 = 𝒙𝒐 , 𝒚𝒐 + 𝒕 𝒂, 𝒃 , 𝒕 ∈ 𝑹 OR 𝒓 = 𝒙𝒐 , 𝒚𝒐 + 𝒕 𝒂, 𝒃 , 𝒕 ∈ 𝑹
NOTE: In the vector equation, t is called a parameter. This means that t can be
replaced by any real number to obtain coordinates of points on the line.
Ex 
Given points, 𝐴(2, −3) and 𝐵(6, −1):
a)
determine the V.E. of the line passing through A and B;
b)
identify two other points on this line.
Ex 
State the V.E. of the line passing through (2,3) that is:
a)
parallel to 𝑟 = 0,1 + 𝑡(−4,5);
_____________________________________
b)
perpendicular to 𝑟 = 0,1 + 𝑡(−4,5) .
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(NOTE:
C.
If 𝑚 = (𝑎, 𝑏), then 𝑚⊥ = (−𝑏, 𝑎) and 𝑚 ∙ 𝑚⊥ = 0.)
PARAMETRIC EQUATIONS of a LINE (R2)
The equation of a line, 𝑥, 𝑦 = 𝑥𝑜 , 𝑦𝑜 + 𝑡 𝑎, 𝑏 , 𝑡 ∈ 𝑅 , can also be written:
PARAMETRIC EQUATIONS of a LINE in R2:
𝒙 = 𝒙𝒐 + 𝒕𝒂
𝒚 = 𝒚𝒐 + 𝒕𝒃
Ex 
𝑡∈𝑅
A line passes through (−2,5) with 𝑚 = (
−1
2
, 3). Determine:
a)
a direction vector with integer components;
b)
the P.E. of the line;
c)
whether the point, (0,6), lies on the line;
d)
the y–intercept of the line.
HOMEWORK: p.433–434 #1–13