Vector Notes:

A quantity with both magnitude and direction is called a vector quantity. Force, velocity,

acceleration and linear displacement are all vector quantities. A quantity with only magnitude is called a

scalar quantity. Mass, length, time and temperature are all examples of scalar quantities. What are

some other examples of vector quantities? Of scalar?

Vectors are easily represented by arrows. The magnitude of the vector is represented by the length of the arrow and will either be labeled or drawn to a specified scale. The direction portion of the vector is represented by the direction the arrow is pointing. There are three commonly terms used to describe direction in vector problems:

bearing is the angle measured from due North, (clockwise)

direction is measured from the positive x-axis, (counterclockwise)

Heading must state one of the following descriptions, North of West, South of East etc…

A force (F1) of 5.66 Newtons is

exerted 45 ° East of North.

Additional Resources:

http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Components

A force (F2) of 4.8 Newtons is exerted

at an angle of 130°.

If a vector v is placed in a Cartesian

coordinate system it can be shown to be the

sum of the horizontal component (X) and

the vertical component (Y) of the vector.

The horizontal displacement can be found

by x2-x1 and the vertical displacement can

be found by y2-y1.

A bird is flying at a velocity of 5 m/s at an

angle of 36.9° North of East. What are the

horizontal and vertical components of this

vector? Solve graphically. Solve

mathematically.

Combining Vectors

Combining vectors can be looked at in two equivalent ways, the triangle method (sometimes

referred to as the tail to tip method) and the parallelogram method. When vectors are drawn to scale,

both of these methods can be used to find the resultant vector graphically.

Triangle Method Parallelogram Method

Vectors can also be combined mathematically. However, before you pick up the calculator, always

draw a sketch of the problem, using one of the methods above.

Example: Tom walks 4 kilometers due east and then turns north and continues walking for 7 additional

kilometers. Calculate Tom’s displacement.

Perpendicular vectors can be combined using the

Pythagorean Theorem.

A2 + B2 = C2

42 + 72 = C2

16 + 49 = C2

65 = C2

√(65) = √(C2)

C = 8.06 Km

Are we finished yet? Why not?

What if vectors fall in exactly the same direction or exact opposite directions? Consider the vector

diagram below. What would the resultant of these vectors look like?

A vector with a magnitude of zero is called a zero vector. Since the initial and terminal points of this

vector coincide, this vector can be thought of as a single point. What would it look like if two vectors

were acting in exactly opposite directions, but did not result in a zero vector?

8.06 Km is the magnitude, but we also need the

direction to make the description of this vector

complete.

Tan θ = 7km/4km

Tan θ = 1.75

θ = 60.3°

(Be sure your calculator is set in degrees and not

radians.)

So the full answer is 8.06 Km 60.3° North of East.

Practice Problems: Find the magnitudes and direction of the resultants for the following pairs of vectors.

1. 12 m/s east and 6 m/s north

2. 8 miles/hour south and 5 miles/hour west

3. 11 N west and 5 N north

4. 3.5 m/s west and 6.5 m/s south

Ans) 1. 13.4 miles 26.6° north of east, 2. 9.4 mile/hour 32° west of south, 3. 12.1 N 24.4° north of west,

4. 7.4 m/s 61.7° south of west

Additional Resources:

Vector practice problems - http://www.physicsclassroom.com/morehelp/vectaddn

Additional problems with solutions - http://www.physicsclassroom.com/calcpad/vecproj/problems

Adding Vectors Game - http://www.physicsclassroom.com/Physics-Interactives/Vectors-and-Projectiles

An airplane is flying with a velocity of 300 Km/hr North.

The airplane encounters a headwind blowing south with a

velocity of 50 Km/hr. What is the resulting velocity of the

plane?

300 Km/hr - 50 Km/hr = 250 Km/hr north

The same airplane on a return flight has a velocity of 300

Km/hr south. The wind continues to blow with a velocity

of 50 Km/hr southward. What is the resulting velocity of

the plane? (This is called a tail wind.)

300 Km/hr + 50 Km/hr = 350 Km/hr south

Non Right-Angle Vectors

So far, we have been combining vectors that form right triangles. How do you find the resultant of

non-right-angle vectors?

One method is to find the horizontal and vertical components of each vector and combine them.

Example: A bird is flying at 4.47 miles/hour at an angle of 26.6° north of east. The bird encounters a

breeze blowing at 2.83 miles/hour at an angle of 45° north of east. Calculate the resulting velocity of

the bird.

Cos 26.6° = x1/4.47

X1 = 4

Sin 26.6° = y1/ 4.47

Y1 = 2

Cos 45° = x2/2.83

X2 = 2

Sin 45° = y2/2.83

Y2 = 2

Xtotal = 4 + 2 = 6

Ytotal = 2 + 2 = 4

A2 + B2 = C2

42 + 62 = C2 , C = 7.21 miles/hour

Tan θ = 4/6, θ = 33.7° north of east

Practice Problems: Find the magnitudes and direction of the resultants for the following pairs of vectors

by combining x and y components.

1. 25N 30° WEST OF NORTH, 50N 15° NORTH OF EAST

2. 20N 45° WEST OF NORTH, 37N 30° WEST OF NORTH

3. 15N 30° NORTH OF EAST, 12N 20° SOUTH OF EAST

4. 17N 65° NORTH OF WEST, 10N NORTH

ANS: 1) 49.78N 44° North of East, 2) 56.56N 54.75° North of West, 3) 24.5N 7.97° North of East,

4) 26.4N 74.2° North of West

Additional Vector Components Resources:

http://www.physicsclassroom.com/class/vectors/Lesson-1/Component-Addition

Vector Notation

Practice Problems:

Sketch the following vectors. Be sure to include components, magnitude and direction.

1. Vector f = 4i - 4j

2. Vector g = (17)

Ans: 1) 2)

A vector can also be thought of as a simple line segment,

having a start point and end point. The vector to the left can

be broken apart into x and y components. In rectangular

vector notation, the x component can be represented as i

and the y component can be represented as j. So, in this

example we could write…

→

d = 3.54i + 3.54j

→

The vector v = -4i - 3j is represented to the left. This can

also be written…

→

V = (−4−3

)

Practice Problems:

Sketch the following pairs of vectors and determine if they are equal, parallel, or opposite.

1. Vector a = 3i = 2j, vector b = -3i - 2j

2. Vector c = 3i + 2j, vector d = 6i + 4j

3. Vector e = 3i + 2j, vector f = (32)

Ans: 1) opposite, 2) parallel, 3) equal

Vector Notation (cont.)

The magnitude of vector v can be represented as…

││ 𝑣→││

It can also be represented as…

│ 𝑣→│

What might the second notation be confused with?

Scalar * Vector

Multiplication can be thought of as repeated addition. For example…

3 * 4 = 4 + 4 + 4 = 12

Now, let’s apply this same thinking to vectors. Consider the vector A…

What do you think 2 *A would look like?

or…

How about -2 * A?

A vector times a scalar results in a vector quantity with the same direction as the original vector but with

a magnitude equal to the product of the scalar term and the magnitude of the original vector.

Using rectangular notation vector A would look like…

A = 2i + 1j

and 2A = (2*2i) + (2*1j) = 4i + 2j

Practice Problems:

Given vector u = -5i + 4j and vector w = 1i - 1j find the following:

1. -7w

2. 3u

3. -1w

Ans.) 1. (−77

), 2) (−1512

) , 3) (−11

)

Scalar Product (Dot procuct)

Take another look at the example above. Change the angle from 60° to 90° and rework the problem.

What is the new solution?

Since the cos 90° is zero, the scalar product of perpendicular vectors will always be zero.

Practice Problem:

Consider the following vectors: vector a = 4i – 4j and vector b = 1i + 7j. Sketch these vectors labeling all

sides and angles of the resultants.

Find the scalar product of these two vectors using a * b = ‖𝑎‖ ‖𝑏‖ cos θ

Find the scalar product of these two vectors using a * b = axbx + ayby

Ans.) Both methods should give an answer of -24

There are several ways in which vectors can be multiplied

together. One such way is called the scalar product. The scalar

product of two vectors is equal to the magnitude of the first

vector times the magnitude of the second vector times the

cosine of the angle between the two vectors.

a * b = ‖𝑎‖ ‖𝑏‖ cos θ

Using the figure to the left, if vector a has a magnitude of 6 and

vector b has a magnitude of 7 then…

a * b = ( 6*7) Cos 60°

a * b = 21

Extra Practice/Resources:

13 page pdf with a variety of vector problems (with answers)

www.mrwaynesclass.com/vectors/vectors.pdf

A variety of vector problems (no answers) organized by type…

http://www.mathworksheetsland.com/topics/vectors.html

Three videos, basic explanation of vectors and finding x and y components…

http://whs.wsd.wednet.edu/faculty/busse/mathhomepage/busseclasses/physics/physics.html

Vector word problems with answers…

http://courseweb.hopkinsschools.org/pluginfile.php/118762/mod_folder/content/0/vector_Homework

_answers/Ch_4_HW_Answers.pdf?forcedownload=1

Solved Samples Problems and Practice Problems with answer key…

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=16&cad=rja&uact=8&ved=0CEoQ

FjAFOAo&url=http%3A%2F%2Ffrontenacss.limestone.on.ca%2Fteachers%2Fdcasey%2F0F7D40F1-

00870BC8.42%2Fvectors%2520worksheet.pdf&ei=lgZNVbneGqHfsASYioGYBg&usg=AFQjCNH4Z3j_iJjnQI

SItBzfUWIOVTdLSQ&sig2=uEStGPlsgbmYfa2UKLSKOA

Vector Vocabulary on Quizlet…

https://quizlet.com/12207337/vector-vocabulary-flash-cards/

Vector Resources from WVU…

http://nextgen.wvnet.edu/Courses/lesson.php?c=6&u=32&l=124&t=Resources

http://nextgen.wvnet.edu/Courses/lesson.php?c=6&u=32&l=123&t=Lesson