OPERATIONS with ALGEBRAIC VECTORS in R2
A.
REPRESENTING VECTORS in R2
y
𝑖 and 𝑗 are defined as unit vectors
pointing along the (+) x–axis and
(+) y–axis respectively
𝒋
0
𝒊
x
𝒊 = (𝟏, 𝟎) and 𝒋 = (𝟎, 𝟏)
Any vector can be expressed using the unit vectors, 𝑖 and 𝑗.
For this reason, vectors 𝑖 and 𝑗 are also called the standard basis vectors in R2.
𝑢 = 𝑂𝑃
Ex.
Ex 
= (𝑎, 𝑏)
component notation
= 𝑎𝑖 + 𝑏𝑗
unit vector notation
If P(2,–4) is a point, then
𝑂𝑃 = (2,–4) is the vector in component form
𝑂𝑃 = 2𝑖 − 4𝑗 is the vector in unit vector form
Write each of the following vectors in the alternate form:
a)
𝑂𝑃 = (6, −5)
b)
𝑢 = −6𝑖 + 3𝑗
c)
𝑢 = (0,8)
d)
𝑂𝑃 = −𝑗
B.
OPERATIONS on VECTORS in COMPONENT FORM
ADDING TWO VECTORS in R2

Given 𝑂𝐴 = (𝑎, 𝑏) and 𝑂𝐷 = (𝑐, 𝑑), determine 𝑂𝐴 + 𝑂𝐷 .
𝑂𝐴 + 𝑂𝐷 =
ADDING TWO VECTORS in R2
To determine the sum of two algebraic vectors,
add their corresponding x– and y– components.
If 𝑂𝐴 = (𝑎, 𝑏) and 𝑂𝐷 = (𝑐, 𝑑), then
𝑶𝑨 + 𝑶𝑫 = (𝒂 + 𝒄, 𝒃 + 𝒅)
Similarly,
𝑶𝑨 − 𝑶𝑫 = (𝒂 − 𝒄, 𝒃 − 𝒅)

MULTIPLYING a VECTOR by a SCALAR in R2
Given 𝑂𝑃 = (𝑎, 𝑏) and a scalar, m, determine 𝑚𝑂𝑃.
𝑚𝑂𝑃 =
MULTIPLYING a VECTOR by a SCALAR in R2
To multiply an algebraic vector by a scalar, multiply
both x– and y– components by the scalar.
If 𝑂𝐴 = (𝑎, 𝑏) and m is a scalar, then
𝒎𝑶𝑷 = (𝒎𝒂, 𝒎𝒃)
Ex 
If 𝑎 = 2,4 and 𝑏 = (−1,3), determine:
a)
𝑎+𝑏
b)
c)
𝑎−𝑏
d)
3𝑎
e)
−2𝑏
f)
2𝑎 − 3𝑏
Ex 
If 𝑥 = 2𝑖 + 3𝑗 and 𝑦 = −𝑖 + 2𝑗, determine:
a)
C.
𝑎+𝑏
𝑥 + 2𝑦
b)
𝑥 + 2𝑦
ALGEBRAIC VECTORS DEFINED BY TWO POINTS
If a vector has initial point A(x1, y1) and
final point B(x2, y2), then
𝑨𝑩 = (𝒙𝟐 − 𝒙𝟏 , 𝒚𝟐 − 𝒚𝟏 )
Ex.
Ex 
If A(5,2) and B(3,4) determine 𝐴𝐵.
Given 𝐴𝐵 = (5, 1) and A(–1, 3) determine the coordinates of the point B.
D.
APPLICATIONS
Ex 
ABC has vertices A(4,7), B(0,4), and C(7,1). Determine its perimeter.
A(4,7)
B(0,4)
Ex 
C(7,1)
Given parallelogram ABCD, determine the coordinates of D.
B(3, –2)
A(–1,8)
C(–5,0)
D(x, y)
Homework: p.324–326 #1, 3–16