VECTORS IN THE PLANE
Section 11-A
Vectors
When an object moves along a straight line, its
velocity can be determined by a single number
that represents both magnitude and direction.
(forward if positive and backward if negative.
While a pair (a, b) determines a point in the
plane, it also determines a directional line
segment or arrow) with its tail at the origin and
head at ( a, b)
Vectors
The length of this arrow represents magnitude,
while the direction in which it points represents
direction. The ordered pair (a, b) represents an
object with both magnitude and direction, called
the position vector.
The vector denoted in
component form is a, b
Vector Notation
• An arrow over a letter
V
– or a letter in
bold face V
• An arrow over two letters
V
A
– The initial and terminal points
– AB or both letters in bold face AB
B
• The magnitude (length) of a vector is notated
with double vertical lines
V
AB
4
Magnitude of a Vector
The magnitude (or absolute value) of v, denoted v
Is the length of the arrow, and the direction of v
is the direction in which the arrow is pointing.
The vector 0  0,0 , called the zero vector has
zero length and no direction.
The magnitude of vector (a,b) is given by
a, b  a 2  b 2
distance formula
Vector Direction
Direction is determined using the angle formed
by the positive x-axis as the initial ray and the
vector as the terminal ray. Angles are measures
in degrees 0    360 or 0    2 radians
Same
magnitude,
different
directions
The direction angle  of a vector v is the angle
formed by the positive half of the x-axis and the ray
along which v lies.
y
y
θ
x
v
θ
v
x
If v = x, y , then tan  =
y
.
x
If v = 3, 4 ,
4
then tan  = and  = 51.13.
3
y
(x, y)
v
x
y
x
7
Equivalent Vectors
• Have both same direction
and same magnitude
(a, b)
Pi  xi , yi 
• Given points Pt  xt , yt 
• The components of a vector
– Ordered pair of terminal point with initial point at (0,0)
–
xt  xi , yt  yi
8
1) Find the magnitude and direction of the

vector v   1, 3
Component form
If a vector in the plane is in standard position

then the component form of v  vx , v y

Is v  a, b where a  v cos  and b  v sin 
A vector with initial point (0, 0) is in standard position and is
represented uniquely by its terminal point (u1, u2).
y
(u1, u2)
x
If v is a vector with initial point P = (p1 , p2) and terminal point
Q = (q1 , q2), then
1. The component form of v is
v = q1  p1, q2  p2
2. The magnitude (or length) of v is
||v|| = (q1  p1 ) 2  (q2  p2 ) 2
y
P (p1, p2) Q (q1, q2)
x
2) Find the component form of the vector with
magnitude 3 and direction Ɵ = 40°
3) Find the component form and magnitude of the
vector v with initial point P = (3, 2) and terminal
point Q = (1, 1).
p1 , p2 = 3, 2
q1 , q2 = 1, 1
So, v1 = 1  3 =  4 and v2 = 1  ( 2) = 3.
Therefore, the component form of v is
v1, v2 =  4, 3
The magnitude of v is
||v|| = (4) 2  (3) 2 = 25 = 5.
Fundamental Vector Operations
Let u = (x1, y1), v = (x2, y2), and let k be a scalar.
1. Scalar multiplication cu = (kx1, ky1)
2. Addition
u + v = (x1+x2, y1+ y2)
3. Subtraction
u  v = (x1  x2, y1  y2)
14
Scalar multiplication is the product of a scalar, or
real number, times a vector.
For example, the scalar 3 times v results in the vector 3v,
three times as long and in the same direction as v.
v
3v
1
The product of - and v gives a vector half as long
2
as and in the opposite direction to v.
v
-
1
v
2
15
Vector Addition
• Sum of two vectors can be represented
geometrically in the parallelogram
representation or the tail-to-head
representation
Note that the
sum of two
vectors is the
diagonal of the
resulting
parallelogram
A
B
Vector Subtraction
• The difference of two vectors is the result of
adding a negative vector
– A – B = A + (-B)
A
B
A-B
-B
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Examples: Given vectors u = (4, 2) and v = (2, 5)
y
-2u = -2(4, 2) = (-8, -4)
(4, 2)
u
(-8, -4)
u + v = (4, 2) + (2, 5) = (6, 7)
y
(2, 5)
v
2u
u  v = (4, 2)  (2, 5) = (2, -3)
y
(2, 5)
(6, 7)
v
(4, 2)
u
x
x
(4, 2)
u
uv
(2, -3)
x
Unit Vectors
• Definition:
– A vector whose magnitude is 1
• Typically we use the horizontal and vertical
unit vectors i and j
– i = <1, 0> j = <0, 1>
– Then use the vector components to express the
vector as a sum
– V = <3,5> = 3i + 5j
19
Unit Vectors
• Definition:
– A vector whose magnitude is 1
• Typically we use the horizontal and vertical
unit vectors i and j
– i = <1, 0> j = <0, 1>
– Then use the vector components to express the
vector as a sum
– V = <3,5> = 3i + 5j
20
4) Find a unit vector for a   5, 2
Angle between two Vectors
The angle between two nonzero vectors is
given by:
 u1v1  u2 v2 

  cos 
 

u
v


1
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5) Find the angle C in triangle ABC determined
by the vertices A( 0, 0), B(3, 5) and C(5, 2)
HOME WORK
Worksheet: Vectors in
a plane