General Physics I, Spring 2011
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Vectors
Vectors: Introduction
• A vector quantity in physics is one that has a magnitude (absolute value) and a direction. We have seen three already: displacement, velocity, acceleration. The value of the vector quantity must be accompanied by its unit. (For example, “The velocity of the car is 3.5 m/s east.”)
• Vectors are represented graphically by arrows. The magnitude of the vector is represented by the length of the arrow. The direction of the vector is indicated by the direction of the arrow.
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direction of the vector is indicated by the direction of the arrow.
• Two vectors are equal if their magnitudes are equal and their directions are the same. The vectors do not have to be in the same place!
• We add, subtract, and multiply vectors according to laws of vector algebra, which are different from those of ordinary algebra.
Vector Addition: Graphical• Vectors are added graphically using the tip-to-tail rule. To add
two vectors, place the tail of the second vector at the tip
(arrowhead) of the first. The sum vector, or resultant, is the
vector whose tail is at the tail of the first vector and whose tip is
at the tip of the second vector. Any number of vectors may be
added using the tip-to-tail rule.
• Vectors may also be added using the parallelogram rule. The
two vectors are placed tip-to-tail as before. Then a vector equal
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two vectors are placed tip-to-tail as before. Then a vector equal
to the first and another equal to second are drawn such that the
four vectors form a parallelogram. The sum or resultant vector
is the diagonal of the parallelogram going from the tail of the
first vector to the tip of the second.
Vector Addition: Tip-to-Tail Rule
44
Vector Addition: Parallelogram Rule
55From: University Physics, 11th edition (Pearson Addison-Wesley)
Negative of a Vector• The negative of a vector is a vector having the same
magnitude but pointing in the opposite direction. The sum of
a vector and its negative must be zero (i.e., the zero vector);
this is easily shown by the tip-to-tail method.
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Vector Subtraction• Subtracting the vector from the vector is
equivalent to adding to . Thus,B�
A�
B−�
A�
( ).A B A B− = + −� �� �
77
Scalar Times a Vector• A scalar quantity in physics is one that is completely specified
by one number and its unit. As you recall, speed is a scalar
quantity; it has only a numerical value (along with its unit).
• A scalar times a vector gives a vector whose magnitude is
the product of the magnitudes of the scalar and the vector. If
the scalar is positive, the product vector points in the same
direction as the original vector. If the scalar is negative, the
product vector points in the opposite direction to the original
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product vector points in the opposite direction to the original
vector.
• Example: The vector has 3.5 times the
magnitude of and points in the opposite direction to .
3.5V A=−��
A�
A�
Scalar Times a Vector
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Two vectors and are added to give a new vector .
Vectors and have equal magnitudes. Nothing else
is known or may be assumed about them. Which
statement(s) must be TRUE?
F�
G�
H�
F�
G�
1. The magnitude of can never be smaller than the magnitude of either or .
H�
F�
G�
H�
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2. The magnitude of can never be greater than the magnitude of either or .
3. The magnitude of always depends on the angle
between and .
4. The magnitude of is always twice the magnitude of
(or ).
H�
F�
G�
H�
F�
G�
H�
F�
G�
Workbook: Chapter 3, Exercises 5 - 8
1111
Vector Components: Introduction• Using the graphical
method for vector algebra
quickly becomes tedious.
Using vector components
allows one to precisely do
vector algebra using
calculations.
• Let us consider motion in
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• Let us consider motion in
two dimensions (2D). An
x-y coordinate system
allows us to specify points
in 2D space. The x-y
coordinate system has 4
quadrants as shown in the
figure.
Component Vectors• Unlike in the one-dimensional
case, there is now an infinite
number of different directions in
which a vector can point. To
specify the direction of a vector,
we can use the angle that the
vector makes with the x-axis or y-
axis.
θ
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axis.
• Consider a vector in 2D space,
which we will describe by an x-y
system. As the picture shows,
there are two vectors, one parallel
to the x-axis ( ) and one
parallel to the y-axis ( ), which
add together to give . These
vectors are called the component
vectors of .
A�
xA�
yA�
A�
A�
The angle specifies
The direction of .
θ
A�
Vector Components• Since a component vector is always along the x or y axis,
we need only its magnitude (absolute value) and sign to
specify it. The magnitude of a component vector along with
a sign (positive or negative) to specify its direction along the
axis is called a component. Since the component is just a
number, its symbol is not written with an arrow over it.
Thus, if the component vector is , the corresponding
component is , which is called the x-component. xA�
xA
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component is , which is called the x-component.
Similarly, the y-component (Ay ) of the vector is the
magnitude of the component vector along with a sign to
indicate direction along the y-axis.
• The next slide has some examples.
xAA�
yA�
Vector Components
1515
Workbook: Chapter 3, Exercises 13 - 15
1616
Calculating Components
Review: Trigonometry
oppositesin .hypotenuse
adjacentcos .hypotenuse
oppositetan .
θ
θ
θ
=
=
=
1717
oppositetan .adjacent
θ =
Calculating Components
• The vector and its component vectors form a right
triangle. The Pythagorean theorem can be used to find the
magnitude of (the hypotenuse) and its direction if the
components are known. The components can be
calculated using trigonometry if the magnitude and
direction are known. We call the process of calculating
components resolving a vector into its components.
A�
A�
1818
Calculating Components• To specify the direction of a vector, we need to find which
quadrant the vector is in. Then we specify the direction as
an angle above or below the positive or negative x-axis.
Quadrant Ax Ay
I + +
II - +θθ
y
Ay
Ax
Ay
Ax
II I
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II - +
III - -
IV + -
θθ
θ
x
θ
AxAx
Ax Ax
Ay Ay
III IV
1tan .y
x
A
Aθ
−=
Calculating Components: Summary
cos
sin .
x
y
A A
A A
θ
θ
=
=
• If the magnitude and direction of a vector are known,
calculate the components using the formulas:
Use quadrants to get sign of the component correct.
• If the components are known, calculate the magnitude and
angle using the following formulas:
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angle using the following formulas:
2 2
1tan
x y
y
x
A A A
A
Aθ
−
= +
=
Specify the direction as θ degrees above or below the positive
or negative x-axis, depending on the quadrant.
Workbook: Chapter 3, Exercises 16 - 18
2121
Adding Vector With Components• If a number of vectors are added then the x-component
of the resultant (sum) vector is simply the sum of the individual x-components. The same rule applies for the ycomponent. The proof is illustrated below for the sum of two vectors.
Components:
x x xC A B= +
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x x x
y y y
C A B
C A B
= +
= +
Textbook: Chapter 3
• Homework Questions and Problems
Q 4; P 11, 13
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