Vectors and Scalars and Their Physical Significance

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

Examples of Scalar Quantities:

Length Area Volume Time Mass

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

Examples of Vector Quantities:

Displacement Velocity Acceleration Force

Vector diagrams are shown using an arrow

The length of the arrow represents its magnitude

The direction of the arrow shows its direction

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

6 N 10 N = 4 N

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

6 m= 10 m

4 m

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

When two vectors are joined tail to tail

Complete the parallelogram The resultant is found by

drawing the diagonal

When two vectors are joined head to tail

Draw the resultant vector by completing the triangle

Solution: Complete the parallelogram (rectangle)

θ

The diagonal of the parallelogram ac represents the resultant force

2004 HL Section B Q5 (a)Two forces are applied to a body, as shown. What is the magnitude

and direction of the resultant force acting on the body?

5 N

12 N

5

12

a

b c

d

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

N 13 512 Magnitude 22

acac

675

12tan

512tan: ofDirection

1

ac

Resultant displacement is 13 N 67º with the 5 N force

13 N

45º5 N

90ºθ

Find the magnitude (correct to two decimal places) and direction of theresultant of the three forces shown below.

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

What is a scalar quantity? Give 2 examples What is a vector quantity? Give 2 examples How are vectors represented? What is the resultant of 2 vector quantities? What is the triangle law? What is the parallelogram law?

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

y v

x

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

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

50 Ny=25 N

x=43.3 N30º

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

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

vy=v Sin θ

x=v Cos θθ

y

Proof:

vxCos

vCosx vySin

vSiny

x

60º

A force of 15 N acts on a box as shown. What is the horizontalcomponent of the force?

Verti

cal

Com

pone

nt

Horizontal Component

Solution:

N 5.76015Component Horizontal Cosx

N 99.126015Component Vertical Siny

15 N

7.5 N12

.99N

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

If a vector of magnitude v has two perpendicular components x and y, and v makes and angle θ with the x component then the magnitude of the components are:

x= v Cos θ y= v Sin θ

vy=v Sin θ

x=v Cosθ

θy

Scalar – physical quantity that is specified in terms of a single real number, or magnitude◦ Ex. Length, temperature, mass, speed

Vector – physical quantity that is specified by both magnitude and direction◦ Ex. Force, velocity, displacement, acceleration

We represent vectors graphically or quantitatively:◦ Graphically: through arrows with the orientation

representing the direction and length representing the magnitude

◦ Quantitatively: A vector r in the Cartesian plane is an ordered pair of real numbers that has the form <a, b>. We write r=<a, b> where a and b are the components of vector v.

Note: Both and r represent vectors, and will be used interchangeably. r

The components a and b are both scalar quantities.

The position vector, or directed line segment from the origin to point P(a,b), is r=<a, b>.

The magnitude of a vector (length) is found by using the Pythagorean theorem:

Note: When finding the magnitude of a vector fixed in space, use the distance formula.

22, babarr

Vector Addition/SubtractionThe sum of two vectors, u=<u1, u2> and v=<v1, v2> is the vectoru+v =<(u1+v1), (u2+v2)>.◦ Ex. If u=<4, 3> and v=<-5, 2>, then u+v=<-1,

5>◦ Similarly, u-v=<4-(-5), 3-2>=<9, 1>

Multiplication of a Vector by ScalarIf u=<u1, u2> and c is a real number, the scalar multiple cu is the vector cu=<cu1, cu2>.◦ Ex. If u=<4, 3> and c=2, then cu=<(2·4), (2·3)>

cu=<8, 6>

A unit vector is a vector of length 1. They are used to specify a direction. By convention, we usually use i, j and k to

represent the unit vectors in the x, y and z directions, respectively (in 3 dimensions).◦ i=<1, 0, 0> points along the positive x-axis◦ j=<0, 1, 0> points along the positive y-axis◦ k=<0, 0, 1> points along the positive z-axis

Unit vectors for various coordinate systems:◦ Cartesian: i, j, and k◦ Cartesian: we may choose a different set of unit

vectors, e.g. we can rotate i, j, and k

◦ To find a unit vector, u, in an arbitrary direction, for example, in the direction of vector a, where a=<a1, a2>, divide the vector by its magnitude (this process is called normalization).

Ex. If a=<3, -4>, then <3/5, -4/5> is a unit vector in the same direction as a.

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21

22

21

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aaaa

au

The dot product of two vectors is the sum of the products of their corresponding components. If a=<a1, a2> and b=<b1, b2>, then a·b= a1b1+a2b2 .◦ Ex. If a=<1,4> and b=<3,8>, then a·b=3+32=35

If θ is the angle between vectors a and b, then

Note: these are just two ways of expressing the dot product

Note that the dot product of two vectors produces a scalar. Therefore it is sometimes called a scalar product.

cosbaba

Convince yourself of the following:

Conclusion: After you define the direction of an arbitrary vector in terms of the Cartesian system, you can find the projection of a different vector onto the arbitrary direction. By dividing the above equation by the magnitude of b, you can find the projection of a in the b direction (and vice versa).

bbonaprojbababa )..(coscos

)..( bonaprojb

ba