Calculus and Vectors – How to get an A+
8.1 Vector and Parametric Equations of a Line in R2
A Vector Equation of a Line in R2
Let consider the line L that passes through the
r
point P0 ( x0 , y0 ) and is parallel to the vector u .
The point P( x, y ) is a generic point on the line.
Ex 1. The vector equation of the line L is given by:
r
L : r = (0,1) + t (−1,2), t ∈ R
a) Find a direction vector for this line.
b) Find a specific point on this line.
c) Find the points A and B on this line corresponding to
t = 1 and t = 4 respectively.
r
P0 P = tu
r
OP − OP0 = tu
r r
r
r − r0 = tu
The vector equation of the line is:
r r
r
r = r0 + tu , t ∈ R
where:
r
Ö r = OP is the position vector of a generic point
P on the line,
r
Ö r0 = OP0 is the position vector of a specific
d) Explain what does represent the equation:
r
r = (0,1) + t (−1,2), t ∈ [1,4]
e) Verify if the points M (−2,5) and N (2,3) are or not on the
line. Hint: Try to find a t corresponding to each point.
point P0 on the line,
r
Ö u is a vector parallel to the line called the
direction vector of the line, and
Ö t is a real number corresponding to the
generic point P .
Note. The vector equation of a line is not unique. It Ex 2. Find two vector equations of the line L that passes
through the points A(2,−3) and B(−1,2) .
depends on the specific point P0 and on the
r
direction vector u that are used.
B Parametric Equations of a Line in R2
Let rewrite the vector equation of a line:
r r
r
r = r0 + tu , t ∈ R
as:
( x, y ) = ( x0 , y0 ) + t (u x , u y ), t ∈ R
Ex 3. Convert each vector equation into the parametric
equations.
r
a) r = (1,−3) + t (−2,5), t ∈ R
Split this vector equation into the parametric
equations of a line in R2:
⎧ x = x0 + tu x
⎨
⎩ y = y0 + tu y
r
b) r = (−2,0) + s (0,−3), s ∈ R
t∈R
8.1 Vector and Parametric Equations of a Line in R
©2010 Iulia & Teodoru Gugoiu - Page 1 of 2
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Calculus and Vectors – How to get an A+
Ex 4. Convert the parametric equations of each
line into a vector equation.
⎧ x = −2 + 3t
t∈R
a) ⎨
⎩y = 5 − t
Ex 5. Find the points of intersection between the line given by
r
L : r = (1,2) + t (−1,1), t ∈ R and the coordinate axes.
⎧ x = 1 − 3s
s∈R
b) ⎨
⎩y = 2
C Parallel Lines
r
Two lines L1 and L2 with direction vectors u1 and
r
u 2 are parallel ( L1 || L2 ) if:
r r
u1 || u 2
or, there exists k ∈ R such that:
r
r
u 2 = ku1
or:
r
r r
u1 × u 2 = 0
or scalar components are proportional:
u2 x u2 y
=
=k
u1x u1 y
r
Ex 6. Consider the line L1 : r = (1,−3) + t (−1,2), t ∈ R . Find the
vector equation of a line L2 , parallel to L1 that passes
through the point M (−1,−13) .
D Perpendicular Lines
r
Two lines L1 and L2 with direction vectors u1 and
r
u 2 are perpendicular ( L1 ⊥ L2 ) if:
r
r
u 2 ⊥ u1
or:
r
r r
u1 ⋅ u 2 = 0
or:
u1x u 2 x + u1 y u 2 y = 0
Ex 7. Show that L1 ⊥ L2 where:
r
L1 : r = (−2,3) + t (2,−1), t ∈ R
E 2D Perpendicular Vectors
r
Given a 2D vector u = (a, b) , two 2D vectors
r
r
r
perpendicular to u are v = (−b, a) and w = (b,−a ) .
r
Ex 8. Consider the line L1 : r = (0,2) + t (2,−3), t ∈ R . Find the
vector equation of a line L2 , perpendicular to L1 that passes
through the point N (−3,0) .
⎧ x = −2s
L2 : ⎨
⎩ y = 5 − 4s
s∈R
Indeed:
r r
r r
u ⋅ v = (a, b) ⋅ (−b, a) = −ab + ba = 0 ⇒ u ⊥ v
F Special Lines
A line parallel to the x-axis has a direction vector
r
in the form u = (u x ,0), u x ≠ 0 .
A line parallel to the y-axis has a direction vector
r
in the form u = (0, u y ), u y ≠ 0 .
Reading: Nelson Textbook, Pages 427-432
Homework: Nelson Textbook: Page 433 #2, 3, 4, 5, 8, 9, 10, 11, 13
8.1 Vector and Parametric Equations of a Line in R
©2010 Iulia & Teodoru Gugoiu - Page 2 of 2
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