Vector
Mathematics
Physics 1
Physical Quantities
A
scalar quantity is expressed in
terms of magnitude (amount) only.
 Common examples include time,
mass, volume, and temperature.
Physical Quantities
A
vector quantity is expressed in
terms of both magnitude and
direction.
 Common examples include velocity,
weight (force), and acceleration.
Representing Vectors
 Vector
quantities can be graphically
represented using arrows.
– magnitude = length of the arrow
– direction = arrowhead
Vectors

All vectors have a head and a tail.
Vector Addition
 Vector
quantities are added
graphically by placing them headto-tail.
Head-to-Tail Method
1.
2.
Draw the first COMPONENT
vector with the proper length and
orientation.
Draw the second COMPONENT
vector with the proper length and
orientation starting from the head
of the first component vector.
Head-to-Tail Method
3.
4.
The RESULTANT (sum) vector is
drawn starting at the tail of the first
component vector and terminating
at the head of the second
component vector.
Measure the length and orientation
of the resultant vector.
To add vectors, move tail
to head and then draw
resultant from original
start to final point.
East
Resultant is
(sqrt(2)) 45◦ south
of East
South
Vector addition is
‘commutative’ (can add
vectors in either order)
East
Resultant is
(sqrt(2)) 45◦ south
of East
South
Vector addition is
‘commutative’ (can add
vectors in either order)
East
South
Resultant is
(sqrt(2)) 45◦ south
of East
South
East
Co-linear vectors make a
longer (or shorter)
vector
Resultant is 3
magnitude South
Co-linear vectors make a
longer (or shorter)
vector
Resultant is 3
magnitude South
North
Can add multiple
vectors.
Just draw ‘head to tail’
for each vector
East
North
East
Resultant is
2 magnitude
45◦ North of East
North
Adding vectors is
commutative.
Resultant is
2 2 magnitude
45◦ North of East
East
East
North
East
South
North
East
North
Equal but opposite
vectors cancel each
other out
West
East
Resultant=0
Resultant is 0.
South
Vector Addition – same direction
A+B=R
A
B
B
A
R=A+B
Vector Addition
• Example: What is the resultant vector of an
object if it moved 5 m east, 5 m south, 5 m
west and 5 m north?
Vector Addition – Opposite direction
(Vector Subtraction)
.
A + (-B) = R
A
B
-B
A
A + (-B) = R
-B
Vectors
• The sum of two or more vectors is called the
resultant.
Practice
Vector Simulator
http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html
Polar Vectors
 Every
vector has a magnitude and
direction
magnitude
angle dire ction
Right Triangles
SOH
CAH
TOA
Vector Resolution
 Every
vector quantity can be
resolved into perpendicular
components.

Rectilinear (component) form of vector:
xy
Vector Resolution
 Vector
A has been resolved into two
perpendicular components, Ax
(horizontal component) and Ay
(vertical component).
A
Ay

Ax
Vector Resolution
 If
these two components were
added together, the resultant would
be equal to vector A.
A
Ay

Ax
Vector Resolution
When resolving a vector graphically, first
construct the horizontal component (Ax).
Then construct the vertical component
(Ay).
 Using right triangle trigonometry,
expressions for determining the
magnitude of each component can be
derived.

Vector Resolution
 Horizontal
A
Component (Ax)
Ay

Ax
Ax
cos 
A
Ax  A  cos
Vector Resolution
 Vertical
A
Component (Ay)
Ay

Ax
sin  
Ay  A  sin 
Ay
A
Drawing Directions

EX: 30° S of W
–
–
Start at west axis and move south 30 °
Degree is the angle between south and west
N
W
E
S
Vector Resolution
 Use
the sign
conventions for
the x-y coordinate
system to
determine the
direction of each
component.
N
(-,+)
(+,+)
E
W
(-,-)
(+,-)
S
Component Method
1.
2.
3.
Resolve all vectors into
horizontal and vertical
components.
Find the sum of all horizontal
components. Express as SX.
Find the sum of all vertical
components. Express as SY.
Component Method
4.
5.
6.
Construct a vector diagram using the
component sums. The resultant of this
sum is vector A + B.
Find the magnitude of the resultant
vector A + B using the Pythagorean
Theorem.
Find the direction of the resultant
vector A + B using the tangent of an
angle .
Adding “Oblique” Vectors

Head to tail method
works, but makes it very
difficult to ‘understand’
the resultant vector
Adding “Oblique” Vectors



Break each vector into
horizontal and vertical
components.
Add co-linear vectors
Add resultant horizontal
and vertical components
Adding “Oblique” Vectors

Break each vector into
horizontal and vertical
components.
Adding “Oblique” Vectors


Break each vector into
horizontal and vertical
components.
Add co-linear vectors
Adding “Oblique” Vectors



Break each vector into
horizontal and vertical
components.
Add co-linear vectors
Add resultant horizontal
and vertical components
Adding “Oblique” Vectors



Break each vector into
horizontal and vertical
components.
Add co-linear vectors
Add resultant horizontal
and vertical components
Using Calculator For Vectors

Can use the “Angle”
button on TI-84
calculator to do
vector mathematics
Using Calculator for Vectors