Vectors An Introduction

There are two kinds of quantities…

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

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

Notating vectors

!  This is how you notate a vector…

!  This is how you draw a vector…

R R

R head tail

Direction of Vectors

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

!  This vector has an angle that is between 0o and 90o.

A x θ

Vector angle ranges

x

y

θ

Quadrant I 0 < θ < 90o

Quadrant II 90o < θ < 180o

θ

Quadrant III 180o < θ < 270o

θ Quadrant IV

270o < θ < 360o

θ

Direction of Vectors

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

would you determine it?

B x θ

Between 180o and 270o

θ or between - 90o and -180o

Magnitude of Vectors

!  The best way to determine the magnitude (or size) 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

Pythagorean Theorem

!  hypotenuse2 = opposite2 + adjacent2 !  c2 = a2 + b2

θ

oppo

site

adjacent

Basic Trigonometry functions

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

θ

oppo

site

adjacent

SOHCAHTOA

Inverse functions

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

θ

oppo

site

adjacent

SOHCAHTOA

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 θ

A

θ 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 θ

A

θ 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

!  θ = tan-1 (Ay/Ax)

x

y

Rx

Ry

θ

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?

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.

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

Practice Problem

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

b)   v = 25 m/s @ -78o

Practice Problem

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

c)  a = 2.23 m/s2 @ 150o

Graphical Addition of Vectors

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

R A + 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 + B b)  Find A + B + C c)  Find A – B.

a)

b)

c)

Vector Addition Laboratory

Vector Addition by Component

Component Addition of Vectors 1)  Resolve each vector into its x- and y-

components. Ax = Acosθ Ay = Asinθ Bx = Bcosθ By = Bsinθ Cx = Ccosθ Cy = Csinθ 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 θ = 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

!  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?