Vectors and
Two-Dimensional
Motion
1
Vector Notation

 When handwritten, use an arrow: A
When printed, will be in bold print: A
 When dealing with just the magnitude
of a vector in print, an italic letter will
be used: A

2
Properties of Vectors

Equality of Two Vectors


Two vectors are equal if they have the
same magnitude and the same direction
Movement of vectors in a diagram

Any vector can be moved parallel to itself
without being affected
3
More Properties of Vectors

Negative Vectors

Two vectors are negative if they have the
same magnitude but are 180° apart
(opposite directions)


A = -B
Resultant Vector

The resultant vector is the sum of a given
set of vectors
4
Adding Vectors
When adding vectors, their directions
must be taken into account
 Units must be the same
 Graphical Methods



Use scale drawings
Algebraic Methods

More convenient
5
Adding Vectors Graphically
(Triangle or Polygon Method)
Choose a scale
 Draw the first vector with the appropriate
length and in the direction specified, with
respect to a coordinate system
 Draw the next vector with the appropriate
length and in the direction specified, with
respect to a coordinate system whose origin
is the end of vector A and parallel to the
coordinate system used for A

6
Graphically Adding Vectors,
cont.



Continue drawing the
vectors “head-to-tail”
The resultant is drawn
from the origin of A to
the end of the last
vector
Measure the length of R
and its angle

Use the scale factor to
convert length to actual
magnitude
7
Graphically Adding Vectors,
cont.
When you have
many vectors, just
keep repeating the
process until all are
included
 The resultant is still
drawn from the
origin of the first
vector to the end of
the last vector

8
Alternative Graphical Method


When you have only
two vectors, you may
use the Parallelogram
Method
All vectors, including
the resultant, are drawn
from a common origin

The remaining sides of
the parallelogram are
sketched to determine
the diagonal, R
9
Notes about Vector Addition

Vectors obey the
Commutative Law
of Addition

The order in which
the vectors are
added doesn’t affect
the result
10
EXAMPLE 3.1 Taking a Trip
A graphical method for finding the resultant
displacement vector
=
+ .
Goal Find the sum of two vectors by
using a graph.
Problem A car travels 20.0 km due
north and then 35.0 km in a direction
60° west of north, as in the figure.
Using a graph, find the magnitude
and direction of a single vector that
gives the net effect of the car's trip.
This vector is called the car's resultant displacement.
Strategy Draw a graph and represent the displacement vectors as
arrows. Graphically locate the vector resulting from the sum of the
two displacement vectors. Measure its length and angle with respect
to the vertical.
SOLUTION
Let represent the first displacement vector, 20.0 km north, and the
second displacement vector, extending west of north. Carefully graph
the two vectors, drawing a resultant vector with its base touching the
base of and extending to the tip of . Measure the length of this vector,
which turns out to be about 48 km. The angle β, measured with a
protractor, is about 39° west of north.
LEARN MORE
Remarks Notice that ordinary arithmetic doesn't work here: the correct
answer of 48 km is not equal to 20.0 km + 35.0 km = 55.0 km!
Question Suppose two vectors are added. The sum of the magnitudes of
the two vectors is equal to the magnitude of the resultant vector
whenever the two vectors are: (Select all that apply.)
at 45° to each other
opposite directions)
direction)
of equal magnitude
perpendicular
anti-parallel (in
parallel (in the same
Vector Subtraction
Special case of
vector addition
 If A – B, then use
A+(-B)
 Continue with
standard vector
addition procedure

13
Multiplying or Dividing a
Vector by a Scalar
The result of the multiplication or division is a
vector
 The magnitude of the vector is multiplied or
divided by the scalar
 If the scalar is positive, the direction of the
result is the same as of the original vector
 If the scalar is negative, the direction of the
result is opposite that of the original vector

14
Components of a Vector
A component is a
part
 It is useful to use
rectangular
components


These are the
projections of the
vector along the xand y-axes
15
Components of a Vector, cont.

The x-component of a vector is the
projection along the x-axis
Ax  A cos 

The y-component of a vector is the
projection along the y-axis
A y  A sin 

Then, A
 Ax  Ay
16
More About Components of a
Vector

The previous equations are valid only if θ is
measured with respect to the x-axis
The components can be positive or negative
and will have the same units as the original
vector
 The components are the legs of the right
triangle whose hypotenuse is A

A A A
2
x

2
y
and
  tan
1
Ay
Ax
May still have to find θ with respect to the positive
17
x-axis
Adding Vectors Algebraically
Choose a coordinate system and sketch
the vectors
 Find the x- and y-components of all the
vectors
 Add all the x-components
 This gives Rx: R x   v x

18
Adding Vectors Algebraically,
cont.

Add all the y-components

This gives Ry: R y 
v
y
Use the Pythagorean Theorem to find
the magnitude of the Resultant:R  R 2x  R 2y
 Use the inverse tangent function to find
the direction of R:
Ry
1
  tan
Rx

19
EXAMPLE 3.2 Help Is on the Way!
Goal Find vector components, given a
magnitude and direction, and vice versa.
Problem (a) Find the horizontal and
vertical components of the 1.00 102 m
displacement of a superhero who flies
from the top of a tall building along the
path shown in Figure a. (b) Suppose
instead the superhero leaps in the other
direction along a displacement vector
to the top of a flagpole where the
displacement components are given by
Bx = -25.0 m and By = 10.0 m. Find the
magnitude and direction of the displacement vector.
Strategy(a) The triangle formed by the displacement and its
components is shown in Figure b. Simple trigonometry gives
the components relative to the standard x-y coordinate
system: Ax = A cos θ and Ay = A sin θ. Note that θ = -30.0°,
negative because it's measured clockwise from the positive xaxis. (b) Apply the proper equations to find the magnitude
and direction of the vector.
SOLUTION
(a) Find the vector components of from its magnitude and direction.
Use the demonstrated equations to find the components of the displacement vector
.
Ax = A cos θ = (1.00 102 m) cos (-30.0°) = +86.6 m
Ay = A sin θ = (1.00 102 m) sin (-30.0°) = -50.0 m
(b) Find the magnitude and direction of the displacement vector from its
components.
Compute the magnitude of from the Pythagorean theorem.
B = √ Bx2 + By2 = √(-25.0 m)2 + (10.0 m)2 = 26.9 m
Calculate the direction of using the inverse tangent, remembering to add 180° to
the answer in your calculator window, because the vector lies in the second
quadrant.
θ = tan
-1
10.0
By
-1
= tan
= -21.8°
-25.0
Bx
( )
θ = 158°
(
)
LEARN MORE
Remarks In part (a), note that cos (-θ) = cos θ; however, sin (-θ) = -sin θ. The
negative sign of Ay reflects the fact that displacement in the y-direction is
downward.
Question What other functions, if any, can be used to find the angle in part (b)?
(Select all that apply.)
None of those listed.
EXAMPLE 3.3 Take a Hike
(a) Hiker's path and the resultant vector. (b) Components of the hiker's total displacement from camp.
Goal Add vectors algebraically and find the resultant vector.
Problem A hiker begins a trip by first walking 25.0 km 45.0° south of east from her
base camp. On the second day she walks 40.0 km in a direction 60.0° north of east,
at which point she discovers a forest ranger's tower. (a) Determine the components
of the hiker's displacements in the first and second days. (b) Determine the
components of the hiker's total displacement for the trip. (c) Find the magnitude and
direction of the displacement from base camp.
Strategy This problem is just an application of vector addition using components.
We denote the displacement vectors on the first and second days by and ,
respectively. Using the camp as the origin of the coordinates, we get the vectors
shown in Figure a. After finding x- and y-components for each vector, we add them
"componentwise." Finally, we determine the magnitude and direction of the
resultant vector , using the Pythagorean theorem and the inverse tangent function.
SOLUTION
(a) Find the components of .
Use the demonstrated equations to find the components of
Ax = A cos (-45.0°) = (25.0 km)(0.707)= 17.7 km
Ay = A sin -(45.0°) = (-25.0 km)(0.707)= -17.7 km
Find the components of .
Bx = B cos 60.0° = (40.0 km)(0.500)= 20.0 km
By = B sin 60.0° = (40.0 km)(0.866)= 34.6 km
.
(b) Find the components of the resultant vector, = + .
To find Rx, add the x-components of
and .
Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km
To find Ry, add the y-components of
and .
Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km
(c) Find the magnitude and direction of .
Use the Pythagorean theorem to get the magnitude.
R = √ Rx2 + Ry2 = √ (37.7 km2 + (16.9 km)2 = 41.3 km
Calculate the direction of using the inverse tangent function.
(
)
θ = tan-1 16.9 km = 24.1°
LEARN MORE
Remarks Figure b shows a sketch of the components of and their directions in space. The
magnitude and direction of the resultant can also be determined from such a sketch.
Question A second hiker follows the same path the first day, but then walks 15.0 km east on the
second day before turning and reaching the ranger's tower. How does the second hiker's
resultant displacement vector compare with that of the first hiker? List all aspects that apply.
(Select all that apply.)
The two displacements have the same direction.
has the same magnitude as the first.
The second hiker's displacement
The second hiker's displacement is greater in magnitude.
The two displacements have different directions.
smaller magnitude than the first.
The second hiker's displacement has a