Vectors

Topic 1 (Cont.)

Lecture Outline

• Vectors and Scalars• Presentation of Vectors• Addition and Subtraction of

vector• Component of Vector

Vectors and Scalars

•A vector has magnitude as well as direction.

•Some vector quantities: displacement, velocity, force, momentum

•A scalar has only a magnitude.

•Some scalar quantities: mass, time, temperature

Presentation of Vectors

• On a diagram, each vector is represented by an arrow

• Arrow pointing in the direction of the vector

• Length of arrow is proportional to the magnitude of the vector

• Symbol or V• Magnitude of the

vector: V or

V

V

Addition of Vectors—Graphical Methods

For vectors in one dimension, simple addition and subtraction are all that is needed.

You do need to be careful about the signs, as the figure indicates.

If the motion is in two dimensions, the situation is somewhat more complicated.

Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.

Adding the vectors in the opposite order gives the same result:

Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

The parallelogram method may also be used; here again the vectors must be tail-to-tip.

Subtraction of Vectors

In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction.

Then we add the negative vector.

Multiplication of a Vector by a Scalar

A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.

V

V

If two vectors are given

such that A + B = 0, what

can you say about the

magnitude and direction

of vectors A and B?

1) same magnitude, but can be in any direction

2) same magnitude, but must be in the same direction

3) different magnitudes, but must be in the same direction

4) same magnitude, but must be in opposite directions

5) different magnitudes, but must be in opposite directions

ConcepTest 3.1aConcepTest 3.1a Vectors IVectors I

If two vectors are given

such that A + B = 0, what

can you say about the

magnitude and direction

of vectors A and B?

1) same magnitude, but can be in any direction

2) same magnitude, but must be in the same direction

3) different magnitudes, but must be in the same direction

4) same magnitude, but must be in opposite directions

5) different magnitudes, but must be in opposite directions

The magnitudes must be the same, but one vector must be pointing in

the opposite direction of the other in order for the sum to come out to

zero. You can prove this with the tip-to-tail method.

ConcepTest 3.1aConcepTest 3.1a Vectors IVectors I

Given that A + B = C, and

that lAl 2 + lBl 2 = lCl 2, how

are vectors A and B

oriented with respect to

each other?

1) they are perpendicular to each other

2) they are parallel and in the same direction

3) they are parallel but in the opposite

direction

4) they are at 45° to each other

5) they can be at any angle to each other

ConcepTest 3.1bConcepTest 3.1b Vectors IIVectors II

Given that A + B = C, and

that lAl 2 + lBl 2 = lCl 2, how

are vectors A and B

oriented with respect to

each other?

1) they are perpendicular to each other

2) they are parallel and in the same direction

3) they are parallel but in the opposite

direction

4) they are at 45° to each other

5) they can be at any angle to each other

Note that the magnitudes of the vectors satisfy the Pythagorean

Theorem. This suggests that they form a right triangle, with vector C as

the hypotenuse. Thus, A and B are the legs of the right triangle and are

therefore perpendicular.

ConcepTest 3.1bConcepTest 3.1b Vectors IIVectors II

Given that A + B = C,

and that lAl + lBl =

lCl , how are vectors

A and B oriented with

respect to each

other?

1) they are perpendicular to each other

2) they are parallel and in the same direction

3) they are parallel but in the opposite direction

4) they are at 45° to each other

5) they can be at any angle to each other

ConcepTest 3.1c ConcepTest 3.1c Vectors IIIVectors III

Given that A + B = C,

and that lAl + lBl =

lCl , how are vectors

A and B oriented with

respect to each

other?

1) they are perpendicular to each other

2) they are parallel and in the same direction

3) they are parallel but in the opposite direction

4) they are at 45° to each other

5) they can be at any angle to each other

The only time vector magnitudes will simply add together is when the

direction does not have to be taken into account (i.e., the direction is the

same for both vectors). In that case, there is no angle between them to

worry about, so vectors A and B must be pointing in the same direction.

ConcepTest 3.1c ConcepTest 3.1c Vectors IIIVectors III

Adding Vectors by Components

Any vector can be expressed as the sum of two other vectors, which are called its components. The process of finding the component is known as resolving the vector into its component.

Because x and y axis is perpendicular, they can be calculate using trigonometric functions.

The components are effectively one-dimensional, so they can be added arithmetically.

Adding vectors:

1. Draw a diagram

2. Choose x and y axes.

3. Resolve each vector into x and y components.

4. Calculate each component using sines and cosines.

5. Add the components in each direction.

6. To find the length and direction of the vector, use:

and .

Example 3-2: Mail carrier’s displacement.

A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office?

Example 3-3: Three short trips.

An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45°) for 440 km; and the third leg is at 53° south of west, for 550 km, as shown. What is the plane’s total displacement?

If each component of a

vector is doubled, what

happens to the angle of

that vector?

1) it doubles

2) it increases, but by less than double

3) it does not change

4) it is reduced by half

5) it decreases, but not as much as half

ConcepTest 3.2 ConcepTest 3.2 Vector Components IVector Components I

If each component of a

vector is doubled, what

happens to the angle of

that vector?

1) it doubles

2) it increases, but by less than double

3) it does not change

4) it is reduced by half

5) it decreases, but not as much as half

The magnitude of the vector clearly doubles if each of its

components is doubled. But the angle of the vector is given by tan

= 2y/2x, which is the same as tan = y/x (the original angle).

Follow-up:Follow-up: If you double one component and not If you double one component and not the other, how would the angle change?the other, how would the angle change?

ConcepTest 3.2 ConcepTest 3.2 Vector Components IVector Components I

ConcepTest 3.3ConcepTest 3.3 Vector AdditionVector Addition

You are adding vectors of length

20 and 40 units. What is the only

possible resultant magnitude that

you can obtain out of the

following choices?

1) 01) 0

2) 182) 18

3) 373) 37

4) 644) 64

5) 1005) 100

ConcepTest 3.3ConcepTest 3.3 Vector AdditionVector Addition

You are adding vectors of length

20 and 40 units. What is the only

possible resultant magnitude that

you can obtain out of the

following choices?

1) 01) 0

2) 182) 18

3) 373) 37

4) 644) 64

5) 1005) 100

The minimumminimum resultant occurs when the vectors

are oppositeopposite, giving 20 units20 units. The maximummaximum

resultant occurs when the vectors are alignedaligned,

giving 60 units60 units. Anything in between is also

possible for angles between 0° and 180°.