Vectors

An Introduction

There are two kinds of quantities…

Vectors are quantities that have both magnitude and direction, such as displacement velocity acceleration

Scalars are quantities that have magnitude only, such as position speed time mass

Notating vectors

This is how you notate a vector…

This is how you draw a vector…

R R

R headtail

Books usually write vector names in bold. You would write the vector name with an arrow on top.

Direction of Vectors

Vector direction is the direction of the arrow, given by an angle.

Draw a vector that has an angle that is between 0o and 90o.

Ax

Vector angle ranges

x

y

Quadrant I0 < < 90o

Quadrant II90o < < 180o

Quadrant III180o < < 270o

Quadrant IV

270o < < 360o

Direction of Vectors

What angle range would this vector have? What would be the exact angle, and how

would you determine it?

Bx

Between 180o and 270o

or between-270o and -180o

Magnitude of Vectors

The best way to determine the magnitude of a vector is to measure its length.

The length of the vector is proportional to the magnitude (or size) of the quantity it represents.

Sample Problem

If vector A represents a displacement of three miles to the north, then what does vector B represent? Vector C?

A

B

C

Equal Vectors

Equal vectors have the same length and direction, and represent the same quantity (such as force or velocity).

Draw several equal vectors.

Inverse Vectors

Inverse vectors have the same length, but opposite direction.

Draw a set of inverse vectors.

A

-A

The Right Triangle

θ

oppo

site

adjacent

hypotenuse

Pythagorean Theorem

hypotenuse2 = opposite2 + adjacent2

θ

oppo

site

adjacent

hypotenuse

Basic Trigonometry functions

sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent

θ

oppo

site

adjacent

hypotenuseSOHCAHTOA

Inverse functions

θ = sin-1(opposite/hypotenuse) θ = cos-1(adjacent/hypotenuse) θ = tan-1(opposite/adjacent)

θ

oppo

site

adjacent

hypotenuseSOHCAHTOA

Sample problem• A surveyor stands on a riverbank directly across the river from a

tree on the opposite bank. She then walks 100 m downstream, and determines that the angle from her new position to the tree on the opposite bank is 50o. How wide is the river, and how far is she from the tree in her new location?

Sample problem• You are standing at the very top of a tower and notice

that in order to see a manhole cover on the ground 50 meters from the base of the tower, you must look down at an angle 75o below the horizontal. If you are 1.80 m tall, how high is the tower?

Vectors: x-component

The x-component of a vector is the “shadow” it casts on the x-axis.

cos θ = adjacent ∕ hypotenuse cos θ = Ax ∕ A

Ax = A cos

A

x

Ax

Vectors: y-component

The y-component of a vector is the “shadow” it casts on the y-axis.

sin θ = opposite ∕ hypotenuse sin θ = Ay ∕ A

Ay = A sin

A

x

y

Ay Ay

Vectors: angle

The angle a vector makes with the x-axis can be determined by the components.

It is calculated by the inverse tangent function

= tan-1 (Ay/Ax)

x

y

Rx

Ry

Vectors: magnitude

The magnitude of a vector can be determined by the components.

It is calculated using the Pythagorean Theorem.

R2 = Rx2 + Ry

2

x

y

Rx

Ry

R

Practice Problem

You are driving up a long inclined road. After 1.5 miles you notice that signs along the roadside indicate that your elevation has increased by 520 feet.

a) What is the angle of the road above the horizontal?

b) How far do you have to drive to gain an additional 150 feet of elevation?

Practice Problem

Find the x- and y-components of the following vectors

a) R = 175 meters @ 95o

b) v = 25 m/s @ -78o

c) a = 2.23 m/s2 @ 150o

Graphical Addition of Vectors

1) Add vectors A and B graphically by drawing them together in a head to tail arrangement.

2) Draw vector A first, and then draw vector B such that its tail is on the head of vector A.

3) Then draw the sum, or resultant vector, by drawing a vector from the tail of A to the head of B.

4) Measure the magnitude and direction of the resultant vector.

A

B

RA + B = R

Practice Graphical Addition

R is called the resultant vector!

B

The Resultant and the Equilibrant

The sum of two or more vectors is called the resultant vector.

The resultant vector can replace the vectors from which it is derived.

The resultant is completely canceled out by adding it to its inverse, which is called the equilibrant.

A

B

R A + B = R

The Equilibrant Vector

The vector -R is called the equilibrant.If you add R and -R you get a null (or zero) vector.

-R

Graphical Subtraction of Vectors

1) Subtract vectors A and B graphically by adding vector A with the inverse of vector B (-B).

2) First draw vector A, then draw -B such that its tail is on the head of vector A.

3) The difference is the vector drawn from the tail of vector A to the head of -B.

A

B

A - B = C

Practice Graphical Subtraction

-B

C

Practice Problem

Vector A points in the +x direction and has a magnitude of 75 m. Vector B has a magnitude of 30 m and has a direction of 30o relative to the x axis. Vector C has a magnitude of 50 m and points in a direction of -60o relative to the x axis.

a) Find A + Bb) Find A + B + Cc) Find A – B.

a)

b)

c)

Vector Addition Lab1. Attach spring scales to force board such that they all have different

readings.2. Slip graph paper between scales and board and carefully trace your

set up. 3. Record readings of all three spring scales.4. Detach scales from board and remove graph paper.5. On top of your tracing, draw a force diagram by constructing vectors

proportional in length to the scale readings. Point the vectors in the direction of the forces they represent. Connect the tails of the vectors to each other in the center of the drawing.

6. On a separate sheet of graph paper, add the three vectors together graphically. Identify your resultant, if any.

7. Did you get a resultant? Did you expect one?8. You must have a separate set of drawings for each member of

your lab group, so work efficiently

Component Addition of Vectors

1) Resolve each vector into its x- and y-components.

Ax = Acos Ay = Asin

Bx = Bcos By = Bsin

Cx = Ccos Cy = Csin etc.

2) Add the x-components (Ax, Bx, etc.) together to get Rx and the y-components (Ay, By, etc.) to get Ry.

Component Addition of Vectors

3) Calculate the magnitude of the resultant with the Pythagorean Theorem (R = Rx

2 + Ry2).

4) Determine the angle with the equation = tan-1 Ry/Rx.

Practice Problem

In a daily prowl through the neighborhood, a cat makes a displacement of 120 m due north, followed by a displacement of 72 m due west. Find the magnitude and displacement required if the cat is to return home.

Practice Problem

If the cat in the previous problem takes 45 minutes to complete the first displacement and 17 minutes to complete the second displacement, what is the magnitude and direction of its average velocity during this 62-minute period of time?

Relative Motion

Relative motion problems are difficult to do unless one applies vector addition concepts.

Define a vector for a swimmer’s velocity relative to the water, and another vector for the velocity of the water relative to the ground. Adding those two vectors will give you the velocity of the swimmer relative to the ground.

Relative Motion

Vs

Vw

Vt = Vs + Vw

Vw

Relative Motion

Vs

Vw

Vt = Vs + Vw

Vw

Relative Motion

Vs

Vw

Vt = Vs + Vw Vw

Practice Problem

You are paddling a canoe in a river that is flowing at 4.0 mph east. You are capable of paddling at 5.0 mph.

a) If you paddle east, what is your velocity relative to the shore?

b) If you paddle west, what is your velocity relative to the shore?

c) You want to paddle straight across the river, from the south to the north.At what angle to you aim your boat relative to the shore? Assume east is 0o.

Practice Problem

You are flying a plane with an airspeed of 400 mph. If you are flying in a region with a 80 mph west wind, what must your heading be to fly due north?