Vectors
A How to Guide
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Scalar vs. Vector
Scalar Quantity
 Numbers only
(magnitude, size)
 Examples: 15
seconds, 200 grams,
5 miles
 States "how much"
 Expressed by a
number and a unit.
Vector Quantity
 Magnitude and
direction.
 Examples: Force (20
N, down), Velocity
(80 meters per
second north)
Distance vs. Displacement
Distance - describes the
change in position of an
object without any
specified direction.
 Scalar quantity.
 Example: If John walks 5
blocks to school, then
realizes he forgot his
homework and walks 5
blocks home to get it.
What total distance has
he walked? _____
10 blocks.
Displacement - describes
the change in position of
an object in a specific
direction.
 Vector quantity.
 Example: In the
homework example, if
school is directly east of
John’s home what is
John's displacement?
______
0 blocks
Speed vs. Velocity
Speed - describes an
object's distance
moved per time. No
direction is specified
 Speed is a scalar.
 Examples: 25 m/s,
90 km/hr, 70 mph
Velocity is speed in a
certain direction.
 Velocity is a vector.
 The magnitude of
velocity is speed.
 Examples: 25 m/s
10°, 90 km/hr NW, 70
mph upward
Vectors
Drawn using an arrow-tipped line.
 The length of the line represents the
magnitude of the vector.
 The direction of the arrow represents the
direction of the quantity.

Vector diagrams

depict the direction and magnitude of a
vector quantity (velocity) by a vector arrow.
Ex: Shows the
velocity of an
object during its
motion

Magnitude is represented by the size of the
vector arrow.
If the size of the arrow is
the same throughout,
then the magnitude is
constant.
Free Body Diagrams

Vector quantities are often represented by
scaled vector diagrams. These are known
as free-body diagrams (fbd).
Drawing Vectors
To draw a diagram correctly,
 clearly list a scale
 draw an arrow in a
specified direction so the
arrow has a head and a
tail
 clearly label the
magnitude and direction
of the vector
1 cm = 1 km
Vector Direction
Vectors can be directed due East, due West, due
South, and due North.
 There are certain rules for identifying a vector
direction when they are not exactly like this.
– The direction of a vector is
often expressed as an angle
of rotation of the vector about
its tail from either east, west,
north or south.
Vector Direction (cont)

For example, a vector can be said to
have a direction of 27 degrees West of
North (meaning a vector pointing
North has been rotated 27 degrees
towards the westerly direction)

A direction of 58 degrees East of South
(meaning a vector pointing South has
been rotated 58 degrees towards the
easterly direction).
Vector Direction (cont.)
Counterclockwise angle of rotation of the
vector from due East.
 A vector with a direction
 A vector with a direction
of 40° is a vector which
of 240 ° is a vector that
has been rotated 40° in
has been rotated 240 °
a counterclockwise
in a counterclockwise
direction relative to due
direction relative to due
East.
East.
Measuring Vectors


Using a protractor and the vectors on your worksheet,
determine the angles appropriately by recording a
degree angle based on a circle of 360°. You will then
list the angle based upon the Cardinal directions of N,
S, E, and W. (eg. 35° N of W or 65° W of N).
Degrees N of . . ., S of . . . E of . . ., W of . . .
Measuring Vectors
A.
________ , __________________ or ____________________.
B.
________ , __________________ or ____________________.
Scaling of a Vector

The magnitude of a vector in a
scaled diagram is depicted by the
length of the arrow.
This diagram shows a vector with a magnitude
of 20 miles. Since the scale is 1 cm = 5
miles, the vector arrow is drawn with a
length of 4 cm. That is 4 cm x (5 miles/cm)
= 20 miles.
Be sure arrows
are drawn in the
proper direction!
VECTOR ADDITION
Steps of Vector Addition:



Start with a bold dot and draw the first
vector. Draw it to scale and in the proper
direction.
Draw the second vector, starting at the end
of the arrow of the first vector (head). Draw
the second vector to scale and in the proper
direction.
The resultant vector is found by connecting
the starting point (tail of the first vector)
with the head (arrow point) of the second
vector.
The resultant vector
(R) is the sum of
two vectors added
together. (R = A + B)
VECTOR ADDITION
Vector Addition
For parallel vectors, add the
magnitudes
Graphically
4mN+1mS=3mN
4 m N + -1 m N = 3 m N
3m East + 4m East = 7m East
A negative vector is in the opposite direction.
4 m N = -4 m S
Vector Practice:
What is the resultant velocity if you are
driving 40 km/h due north while caught in
hurricane Charlie where the wind is
blowing due north at 20 km/h.
A hiker walks 56 km due west, then turns
around and walks 25 km east. What is
the hiker's displacement? (Remember,
draw the resultant from the starting point
to the head of the last vector.)
Vector Addition

Independent of order
Vector Subtraction
Halliday, Wiley Publishing, Physics 6th
ed

 b is the same magnitude
as b but in the opposite
direction
So, subtraction is the same
as adding a – vector.
   

d  a  b  a  ( b )
Halliday, Wiley Publishing, Physics 6th ed
Halliday, Wiley Publishing, Physics 6th ed
D = A - B = A + (-B)
Vector Multiplication

Multiplying a vector (a ) by a scalar (s) results
in a new vector where

The magnitude is the product of the vector times
the absolute value of the

scalar.
a s
The direction is the same

direction of a if s is positive,
but the opposite if s is
negative.

Vector Division

Dividing a vector (a ) by
 a scalar (s) is the
same as multiplying a by 1/s.
The magnitude is the product of the vector times
the absolute value of the
 1
scalar.
a
s
 The direction is the same

direction of a if s is positive,
but the opposite if s is
negative.

Clicker Understanding

Which of the vectors below best
represents the vector sum 𝑃+𝑄?
Clicker Understanding

Which of the vectors below best
represents the vector sum 𝑃 - 𝑄?
Clicker Understanding

Which of the vectors below best
represents the vector sum 𝑄 - 𝑃?
90° Vectors Addition
The Pythagorean Theorem is useful to
determine the resultant when adding only
two vectors that are at right angles to
each other.
Magnitude Calculation

A hiker leaves camp and hikes 11 km,
north and then hikes 11 km, east.
Determine the resulting displacement of
the hiker.
A2 + B2 = C2  112 + 112 = C2  242 = C2  242  C 2
C = 15.6 km
Direction Calculation

SOH CAH TOA is another useful tool for
90° vectors.
Direction Calculations

This can be used to find any of the angles
in a problem. Use the hiker example to
find all the angles.
opposite
11

 .7051
sin  
hypotenuse 15.6
  sin 1 (0.7051)  45
A Different Challenge
A jogger runs 10 km West and then 5 km South.
What direction does he end up relative to his
starting point.
opposite
5km
 tan  
 0.5
adjacent
10km
  tan (0.5)  26.6 
1
Resolving the vector

If the magnitude and direction of a vector
are known, it is possible to find the
components of the vector. This is called
“resolving the vector into its
components.”
Vector Components

What is a component?
– In two dimensions, the vector components of a
vector are two perpendicular vectors Ax and Ay
that are parallel to the x and y axes, respectively,
and add together
so that
 vectorially


A  Ax  A y
Vector Components

This process can be carried out with the aid of
trigonometry, because the two perpendicular
vector components and the original vector form
a right triangle.
VECTOR RESOLUTION
1.
Construct a sketch of the
vector in the indicated
direction; label its
magnitude and the angle
which it makes with the
horizontal.

Example: If a dog is
restrained by a leash
exerting a 60 N force at a
40°, what are the
horizontal and vertical
components of the force?
VECTOR RESOLUTION
•


2. Draw a rectangle about
the vector so the vector is
the diagonal of the
rectangle; beginning at the
tail, sketch vertical and
horizontal lines, then do
the same for the head.
The result will form a
parallelogram.
3. Draw the components
and place arrowheads to
indicate direction.
4. Label the components

Example:
Fvertical
Fhorizontal
VECTOR RESOLUTION

5. Trigonometry:
– Use sine to
determine the
length of the side
opposite the angle.
– Use cosine to
determine the
adjacent side.
46.0 N
46.0
Vectors in Space .

The landing speed of the space shuttle Atlantis is
347 km/h. If the shuttle is landing at an angle of
15 below the horizontal, what are the horizontal
and vertical components of its velocity?
Clicker Understanding

What are the xand y-components
of these vectors?
A. 3, 2
B. 2, 3
C. -3, 2
D. 2, -3
E. -3, -2
Clicker Understanding
The following vectors
have length 4.0 units.
What are the x- and
y-components of
these vectors?
A. 3.5, 2.0
B. -2.0, 3.5
C. -3.5, 2.0
D. 2.0, -3.5
E. -3.5, -2.0
Clicker Understanding
The following vectors
have length 4.0 units.
What are the x- and
y-components of
these vectors?
A. 3.5, 2.0
B. 2.0, 3.5
C. -3.5, 2.0
D. 2.0, -3.5
E. -3.5, -2.0
Clicker Understanding

The diagram shows
two successive
positions of a particle;
it’s a segment of a full
motion diagram.
Which of the
acceleration vectors
best represents the
acceleration between
vi and vf?
Boat Problems
Adding Vectors with Components
Magnitude: Cx=Ax + Bx; Cy=Ay + By
Using Pythagorean's Theory:
C 2  C x2  C y2
C  C x2  C y2
Adding Vectors with Components
1. Resolve each vector
2. Add all the horizontal
3. Calculate the
into its horizontal (x) and vector components together resultant’s magnitude by
vertical (y) components. and then all the vertical
C  C x2  C y2
vector components together.
and direction by:
Ax  A cos 
(Cx=Ax + Bx and Cy=Ay + By)
Ay  A sin 
 Cy
  tan 
 Cx
1



Golf Shot

A golfer, putting on a green, requires
three strokes to “hole the ball.” During
the first putt, the ball rolls 5.0 m due east.
For the second putt, the ball travels 2.1 m
at an angle of 20.0° north of east. The
third putt is 0.50 m due north. What
displacement would have been needed to
“hole the ball on the very first putt?
Adding Multiple (>2) Vectors
Rotating Coordinate Systems
Halliday, Wiley Publishing, Physics 6h ed
Rotating the coordinate
system xy of (a) an angle of
Φ results in coordinate
system x’y’ of (b)
Where,
 Magnitude, a
a  a x2  a y2  a x 2  a y 2

Direction, θ
   