General Physics I, Spring 2011

11

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

99

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�

1010

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

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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�

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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

1919

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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