V E C T O RS

VECTORSVECTORS

Why Study Vectors?

1. Vectors are the best way to show both mathematically and graphically the complex relationsips between appropriate physical quantities and concepts.

2. Vector representations are concise, explicit and accurate.

3. Vector rules are uniform and consistent across vector operations, ensuring correctness of the description of physical process.

Vectors are mathematical quantities fully

described by both a magnitude and a direction.

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Because direction is an important characteristic of a vector, arrows are used to represent them.

The direction that the arrow is pointing represents the direction of the vector.

The length of the arrow is proportional to themagnitude of the vector

2,3,4

The tip of the arrow, that is, its point, is called the head of the vector.

The other end of the arrow is called the tail of the vector.

5,6

Vectors are usually drawn to scale.

The green arrow is twice as long as the red arrow,indicating that it has twice the magnitude.

If these were velocity vectors and the green vectorrepresented a velocity of 10 m/s in the positive x direction,then the red vector would be interpreted as 5 m/s in thePositive x direction.

10 m/s

5 m/s

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Vector Notation

In physics textbooks, the letter may be simply in a bold font with no arrow.

In physics, a vector is usually named with a single letter with an arrow above it.

The vector has a magnitude of 2 km and a direction 30° North of East.

If the car in the diagram had moved 4 km instead of 2 km, the arrow would have been drawn twice as long.

The practice of using the length of an arrow to represent the magnitude of a vector applies to any kind of vector.

Vector Quantities

DisplacementDirection

VelocityAcceleration

Momentum Force

Weight Drag

Lift Thrust 8

The direction of a vector is often stated in terms of North, South, East, and West.

Note: The positive or negative on an arrow only indicates its direction,such as forward (+), backward (-) or upward (+),downward (-). Forcesare neither positive or negative.

A Cartesian coordinate system is typically used for this purpose. It consists of a pair of lines on a flat surface or plane, that intersect at right angles. 

For the direction of the arrow to be meaningful, some sort of coordinate system is necessary

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In the figure above point P1 has coordinates (3, 4), and point

P2 has coordinates (-1, -3).

(x,y)

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Examples

Note: The positive or negative on an arrow only indicates its direction, such as forward (+), backward (-) or upward (+),downward (-). Vectors are neither positive nor negative. The Physics Classroom

Conventions for Describing Directions of Vectors

We’ve already seen that vectors can be directed due East, due West, due South, and due North.

But some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more north than east.

Thus, there is a clear need for some form of a convention for identifying the direction of a vector that is not due East, due West, due South, or due North.

In cases where the direction of a vector that is not dueEast, due West, due South, or due North, the direction ofa vector can be represented as degrees.

The Physics Classroom

11

The direction of a vector can be expressed as a counterclockwise angle of rotation of the vector about its “tail” from due East.

Using this convention, a vector with a direction of 40 degrees is a vector that has been rotated 40 degrees in a counterclockwise direction relative to due east.

A vector with a direction of 240 degrees is a vector that has been rotated 240 degrees in a counterclockwise direction relative to due east.

Representing the Magnitude of a Vector

The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale.

For example, the diagram below shows a vector with a magnitude of 20 miles.

Since the scale used for constructing the diagram is 1 cm = 5 miles, the vector arrow is drawn with a length of 4 cm.

That is, 4 cm x (5 miles/1 cm) = 20 miles.

Based on the scale, what length would you draw thevector shown above?

5 cm

Question?

During a relay race, runner A runs a given distance due north and then hands off the baton to runner B, who runs for the same distance in a south-easterly direction.

Are they equal?

No, they are not equal!

In order for two vectors to be equal, the magnitude anddirection of both vectors must be the same.

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Equivalent VectorsThese two vectors are equivalent. They have the same length and direction.

You can move vectors around on the coordinate system. So long as you do not change their length or orientation they are equivalent.

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A vector having the same magnitude but opposite direction to a vector A, is -A.

These two vectors are not equal. Even though their magnitudes appear to be the same, their directions are not the same.

(a) The displacement vector for a woman climbing 1.2 m up a ladder is represented by the letter D.

(b) The displacement vector for a woman climbing 1.2 m down a ladder is represented by the letter -D

(a)The force vector for a man pushing on a car with 450N of force in a direction due east is F.

(b)The force vector for a man pushing a car with 450N of force in a direction due west is –F.

Note: The positive or negative on an arrow only indicates its direction, such as forward (+), backward (-) or upward (+),downward (-). Forces are neither positive or negative.

P and Q are two vectors represented in the diagram below.

                       

P and Q are represented as two vectors having equal magnitudes but opposite directions therefore, |P| = |Q|.

Addition of Collinear Vectors

Points are said to be collinear if they lie on a single straight line, especially if it is related to a geometric figure such as a triangle.

We are thus going to learn to add vectors that arein a straight line.

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The order in which you add vectors does not matter.

Addition of vectors:

Two or more vectors may be added together to produce their sum.

If two vectors have the same direction, their resultant has a magnitude equal to the sum of their magnitudes and will also have the same direction.

15, 16

(a) The man is pushing on the desk with a force of 40 N to the right. The boy is pushing on the desk in the same direction with a force of 20 N.

(b) The total force the man and the boy are applying to the desk to the right is shown by the vector b. Since both force vectors are in the same direction, they are added

Subtraction of vectors:

Two or more vectors may be subtracted to produce their difference.

If two vectors have opposite direction, their resultant has a magnitude equal to the difference of their magnitudes. The direction will be determined by the magnitude of the largest vector.

17, 18

(a) The man is pushing on the desk with a force of 40 N to the right. The boy is pushing on the desk in the same direction with a force of 20 N.

(b)The girl is pushing in the opposite direction with a force of 10 N

(c) The difference of the vectors is 50 N.

I walk 100 m north and then 200 m south. What is my total displacement from my starting point?

100 m N – 200 m S =-100 m South

In the example to the left,the two vectors are parallel.

How would you add or subtractvectors that are perpendicular?

In the diagram below, two perpendicular vectors are represented by two sides of a triangle in sequence.

In sequence means that the vectors are placed such that the tail of vector BC begins at the arrow head of the vector placed before it, AB.

The third closing side of the triangle (AC), represents the sum (or resultant ) of the two vectors in both magnitude and direction.

Triangle law of vector addition

In this case,the closing side of the right triangle AC represents the sum (i.e. resultant) of individual displacements AB and BC.

Draw the resultant vector by connecting the tail of the first vector to the head of the last vector..

AC = AB + BC

The direction of the resultant vector must be included. The vector’s direction is show as an angle, measured from the horizontal.

If drawn to scale, a ruler is used to measure the magnitude of the resultant vector and a protractor is used to measure the angle (direction) of the resultant vector.

The triangle law does not restrict with which vector or where to start.

Also, it does not put conditions with regard to any specific direction for the sequence of vectors, like clockwise or counter clockwise, to be maintained.

In figure (i), the law is applied starting with vector,b.

In figure (ii) the law is applied starting with vector,a.

In either case, the resultant vector, c, is same in magnitude and direction.

The Physics Classroom

Tail Wind Head Wind Cross Wind

These vectors are notparallel, they areperpendicular.

If two vectors areat right angles to eachother, to get their sum,you need to findthe resultant.

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The resultant vector is the vector that “results” fromcombining two or more vectors together.

The resultant is the diagonal of a rectangle constructedwith the two vectors as sides.

20, 21

Step technique to find the resultant of a pair of vectorsthat are at right angles to each other.

Draw the twovectors withtheir tailstouching.

Draw a parallelprojection ofeach vector with dashedlines to form arectangle.

Draw the diagonalfrom the pointwhere the two tails are touching.

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1. Draw the twovectors withtheir tailstouching.

2. Draw a parallel projection of each vector with dashed lines to form a rectangle.

3. Draw the diagonal from the point where the two tails are touching.

Which vector shown below is the resultant?

b is the resultant. Components will havetheir tails touching.

The equation is a + c = b

Now that we have drawn the resultant, how dowe figure out the new speed?

90°

If two vectors are at right angles to each other, we can use the PythagoreanTheorem (a2 + b2 = c2) tosolve for the resultant.

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Resultant2 = (60 km/h)2 + (80 km/h)2 =

= 3600 (km/h)2 + 6400 (km/h)2

= (km/h)2

= 100 km.h

a2 + b2 = c2

40

30

I walked 40 km east then 30 km north. What's my final displacement?

We first walk 40 km east. Draw the vector.From the tip of the oldarrow, draw the 30km north. To find the final displacement, draw an arrow from the end of the first vector to the tip of the last vector.

Using the Pythagorean Theorem, our total displacement is 50 km NE.

d

(40)2 + (30)2 = d2

1600 + 900 = d2

2500 = d2

= d50 km North of East = d

Let’s Practice

R2 = (5)2 + (10)2

R2 = 125R = SQRT (125)R = 11.2 km

R2 = (30)2 + (40)2

R2 = 2500R = SQRT (2500)R = 50 km

For any square, the length of the diagonal is the square route of two, ( ) or 1.414,times either of the sides.

Special Case

If the rectangle formed is a square, we use the as the resultant.

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1.414 x 100 = 141.4

For any square, the length of the diagonal is , or 1.414, times either of the sides.

Let’s Practice

1.414 x 75 = 106.05

A plane can travel with a speed of 80 mi/hr with respect to the air. Determine the resultant velocity of the plane (magnitude only) if it encounters a

1. 10 mi/hr headwind

2. 10 mi/hr tailwind

3. 10 mi/hr crosswind

4. 60 mi/hr crosswind

70 mi/hr

90 mi/hr

80.6 mi/hr

100 mi/hr

1. A hiker walked walked 3.47 km [E] and 5.32 km [N]. What distance (in km) did the hiker walk?

2. What was the magnitude (in km) of the hiker's displacement in the previous question?

8.79 km

6.35 km

Vector representation review:

1. The reference direction is indicated.

2. Vectors are drawn to scale and the scale is indicated.

3. The vectors are represented as arrows with a length proportional to their magnitude and are correctly orientated with respect to the reference direction.

4. The direction of the vector is indicated by an arrowhead.

5. The arrows should be labeled to show which vectors they represent.

Part 2 Vector Components

Any vector directed in two dimensions can be thought of as having two parts.

Each part of a two-dimensional vector is known as a component.

In this case, two dimensional means that it has both an upward and a Rightward component.

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The process of determining the components ofa vector is called resolution.

Any vector drawn on a piece of paper can beresolved into vertical and horizontal components.

The Physics Classroom

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If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components - an upward component and a rightward component.

The Physics Classroom

The upward and rightward force on the chain is equivalentto an upward force and a rightward force by two chains.

The Physics Classroom

2. A rectangle is drawn that encloses the vector V as its diagonal.

V

• Vertical lines are drawnfrom the tail of the vector (top)

V

Steps to resolve a vector into components!

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Component x

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Sources:The Physics ClassroomConceptual Physics – Paul HewittPhysics – Cutnell and Johnson, third edition

The term “sequence” means that the vectors are placed such that the tail of a vector begins at the arrow head of the vector placed before it.