Vectors and Scalars

Vectors and Scalars

Many quantities in geometry and physics, such as area, volume, temperature, mass, and time, can be characterized as a single real number scaled to appropriate units of measure. These are called scalar quantities, and the real number associated with each is called a scalar.

Other quantities, such as force, velocity, and acceleration, involve both magnitude and direction and cannot be characterized completely by a single real number. A directed line segment is used to represent such a quantity. A directed line segment has an initial point and terminal point for example PQ would have initial point P and terminal point Q, and its length or (magnitude) is denoted as ||PQ|| or sometimes as |PQ|. The set of all directed line segments that are equivalent to a given directed line segment are vectors in the plane. Vectors are denoted in typesets as boldface letters such as u, v, and w. Written by hand they are usually denoted as, , andv u w

A scalar quantity is a quantity that has magnitude only and has no direction in space

Scalars

Examples of Scalar Quantities:

Length Area Volume Time Mass

A vector quantity is a quantity that has both magnitude and a direction in space

Vectors

Examples of Vector Quantities: Displacement Velocity (speed in a given direction) Acceleration (rate of which something speeds

up or slows down in a direction) Force (gravity for example, or pushing an

object with a force of 10 Newtons (N)).

Vector diagrams are shown using an arrow

The length of the arrow represents its magnitude

The direction of the arrow shows its direction

Vector Diagrams

u

v

Because these two vectors (u and v) are going in the same direction and have the same magnitude (length) then the vectors are the same u=v.

-u

Because these two vectors are the same length but are in opposite directions we have u and –u.

Vectors in opposite directions:6 m s-1 10 m s-1 = 4 m s-1

6 N 10 N = 4 N

Resultant of Two Vectors

Vectors in the same direction:6 N 4 N = 10 N

6 m= 10 m4 m

The resultant is the sum or the combined effect of two vector quantities

The sum of two vectors can be represented geometrically by positioning the vectors (without changing their magnitudes or directions) so that the initial point of one coincides with the terminal point of the other. The vector u+v is called the resultant vector.

u

vu+v

If you have a vector v and then –v is a vector of the same magnitude, however it is directed in the opposite direction. This is a property what we will use to subtract two vectors such as u-v, we will change to addition of u + (-v) so you need to locate vector v, change its direction and then move it to tip-tail.

u

v

-v

Animation done by Taylor Cox (the guy)

u-v

The Parallelogram Law When two vectors are joined

tail to tail Complete the parallelogram The resultant is found by

drawing the diagonal

The Triangle Law When two vectors are joined

head to tail Draw the resultant vector by

completing the triangle

Suppose that we have the following two vectors

Adding Vectors “Tip to Tail” method

u

v

Graphically represent u+v.

uv

u+v

Resultant vector

Graphically represent 2u+3v

uv

2u

3v

Is that the same thing as v+(-u)?

Represent v-u

uv

-u

v-u

Sketch u+v, 2(u+v), u-v

u

v

Vector Algebra

-component form-vector addition

-vector subtraction-scalar multiplication

• Component form of a vector is essentially working backwards to identify the two perpendicular vectors that would add (tip to tail) to provide you with the current vector. These two perpendicular vectors are essentially the x-direction and y-direction distances. These values are referred to as the COMPONENTS of your particular vector.

32

v

The x component of v is 3The y component of v is 2

Thus we write v = (3,2) or

This is referred to as COMPONENT FORM of v

(3, 2)v

• You can find the component form of a vector by subtracting the ordered pairs of tip and tail.

v (x2,y2)

(x1,y1)

x2-x1

y2-y1

v (4,3)

(1,1)

x2-x1

y2-y1 = 3-1 =2

=4-1=3

If we needed to find the length of v we would use the Pythagorean Theorem, and then the Pythagorean Theorem would lead us directly to the distance formula. Because we are talking about length it will be positive so we use notation that implies absolute value. This is the equation for the size or MAGNITUDE of the vector.

2 22 1 2 1| | ( ) ( )AB x x y y

General Example

Find the magnitude of the following

•

• Vector addition (algebraically) is quite simple if your vectors are both expressed in component form.

• Suppose we have two vectors v and w. In component form v=(a,b) and w=(c,d) then vector addition states that v+w=(a,b)+(c,d), which means v+w = (a+c,b+d)

Example

• If v=(1,2) and w=(5,1) find v+w.• v+w = (1,2)+(5,1) = (1+5, 2+1) = (6,3)• v+w = (6,3).• Think of it graphically/geometrically. The vectors can start anywhere in the

Cartesian Plane, lets assume that one starts at the origin, and the other starts at the tip of the first (you can assume that they both start at the origin and then use parallelogram technique to verify your answer)

v

w

1

25

1Resultant vector v+w is purpleIdentify its components. Are they (6,3)?

Vector Subtraction• Vector subtraction is the same concept ifv=(a,b) and w=(c,d) then v-w=(a-c, b-d)

Remember geometrically vector subtraction is the same as addition however you add the opposite direction of the vector. If I want to subtract v-w, geometrically we would take v + (-w). And the (-w) would be the same exact vector as w but going in the opposite direction.

v

w

Scalar Multiplication

• Refers to the idea that we can take any vector in component form and multiply it by any constant that we want to. In light of doing so, the components are both changed by the same scalar.

• If v=(3,2) and we want 3v then we multiply both components by 3, thus 3v=(9,6)

• The constant 3 is our SCALAR

Summary of Algebraic Operations on Vectors

1 1 2 2

1 2 1 2

1 2 1 2

1 1

, , ,

,

,

, ,

if x y and x y then

x x y y

x x y y

c cx cy c

u v

u v

u v

u

Properties of Vectors

• u+v = v+u (commutative)• u + (v + w) = (u + v) + w (associative)• u + 0 = u (identity of addition)• u + (-u) = 0 (subtraction)

Find the magnitude of the resultant vector8 l

b

14 lb

39o

Copyright © 2011 Pearson Education, Inc. Slide 10.3-27

10.3 Algebraic Interpretation of Vectors

• A vector with its initial point at the origin is called a position vector.

• A position vector u with endpoint (a,b) is written as u = a, b, where a is called the horizontal component and b is called the vertical component of u.


10.3 Finding Horizontal and Vertical Components

.

Horizontal and Vertical ComponentsThe horizontal and vertical components, respectively, of a vector u having magnitude |u| and direction angle are given by

a = |u| cos and b = |u| sin .

That is, u = a, b = |u| cos , |u| sin .

When resolving a vector into components we are doing the opposite to finding the resultant

We usually resolve a vector into components that are perpendicular to each other

Resolving a Vector Into Perpendicular Components

y v

x

Here a vector v is resolved into an x component and a y component

If a vector of magnitude v and makes an angle θ with the horizontal then the magnitude of the components are:

x = v Cos θ y = v Sin θ

Calculating the Magnitude of the Perpendicular Components

vy=v Sin θ

x=v Cos θθ

y

Proof:

vxCos

vCosx vySin

vSiny

x


10.3 Finding Horizontal and Vertical Components

Example From the figure, the horizontal component isa = 25.0 cos 41.7° 18.7. The vertical component isb = 25.0 sin 41.7° 16.6.

60º

2002 HL Sample Paper Section B Q5 (a)A force of 15 N acts on a box as shown. What is the

horizontalcomponent of the force?

Problem: Calculating the magnitude of perpendicular components

Verti

cal

Com

pone

nt

Horizontal Component

Solution:

N 5.76015Component Horizontal Cosx

N 99.126015Component Vertical Siny

15 N

7.5 N

12.9

9 N

Solution:

Problem: Resultant of 2 Vectors

Complete the parallelogram (rectangle)

θ

The diagonal of the parallelogram ac represents the resultant force

Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? A forces of 12N due East and 5N due South

5 N

12 N

5

12

a

b c

d

The magnitude of the resultant is found using Pythagorean Theorem on the triangle abc

N 13 512 Magnitude 22

acac

1

5Direction of : tan12

12tan 22.65

ac

13 N

45º5 N

90ºθ

Find the magnitude (correct to two decimal places) and direction of the

resultant of the three forces shown below.

Problem: Resultant of 3 Vectors

5 N

5

5

Solution: Find the resultant of the two 5 N forces first (do right angles first)

a b

cdN 07.75055 22 ac

45155tan

7.07 N

10 N

135º

Now find the resultant of the 10 N and 7.07 N forces

The 2 forces are in a straight line (45º + 135º = 180º) and in opposite directions

So, Resultant = 10 N – 7.07 N = 2.93 N in the direction of the 10 N force

2.93 N

Here we see a table being pulled by a force of 50 N at a 30º angle to the horizontal

When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N

Practical Applications

50 Ny=25 N

x=43.3 N30º

We can see that it would be more efficient to pull the table with a diagonal force of 50 N

A person in a wheelchair is moving up a ramp at constant speed. Their total weight is 900 N. The ramp makes an angle of 10º with the horizontal. Calculate the force required to keep the wheelchair moving at constant speed up the ramp. (You may ignore the effects of friction).

Solution:

If the wheelchair is moving at constant speed (no acceleration), then the force that moves it up the ramp must be the same as the component of it’s weight parallel to the ramp.

10º

10º80º

900 N

Complete the parallelogram.Component of weight

parallel to ramp: N 28.15610900 Sin

Component of weight perpendicular to ramp:

N 33.88610900 Cos

156.28 N

886.33 N

Inclined Planes


10.3 Finding the Magnitude of a Resultant

Example Two forces of 15 and 22 newtons act on a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force.

Solution From the figure, the anglesof the parallelogram adjacent to angle Peach measure 80º, since they are supplementary to angle P. The resultantforce divides the parallelogram into twotriangles. Use the law of cosines on either triangle.

|v|2 = 152 + 222 –2(15)(22) cos 80º |v| 24 newtons


10.3 The Unit Vector

• A unit vector is a vector that has magnitude 1. • Two very useful unit vectors

are defined asi = 1, 0 and j = 0, 1.

i, j Forms for Unit VectorsIf v = a, b, then v = ai + bj.


10.3 Dot Product

Example Find the dot product 2, 3 · 4, –1.

Solution 2, 3 · 4, –1 = 2(4) + 3(–1)=

8 – 3=

5

Dot Product

The dot product of two vectors u = a, b and v = c, d is denoted u · v, read “u dot v,” and given by

u · v = ac + bd.


10.3 Applying Vectors to a Navigation Problem

Example A plane with an airspeed of 192 mph is headed on abearing of 121º. A north wind is blowing (from north to south) at 15.9 mph. Find the groundspeed and the actual bearing of the plane.

Solution Let |x| be groundspeed. Wemust find angle . Angle AOC = 121º.Find |x| using the law of cosines .

mph 7.200

121cos)9.15)(192(29.15192

261,40

222

x

x

89.306792320.sin7.200

121sin9.15

sin


10.3 Finding a Required Force

Example Find the force required to pull a wagon weighing 50 lbs up a ramp inclined at 20º to the horizontal.(Assume no friction.)

Solution The vertical 50 lb force BA represents the force of gravity. BA is thesum of the vectors BC and –AC. VectorBC represents the force with which the weight pushes against the ramp. Vector BF represents the force required to pull the weight up the ramp. Since BF and AC are equal, | AC | gives the magnitude of the required force.

sin20 50sin20 17.1 lbs AC AC50

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Vectors and Scalars

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