Lecture Notes 1-SMES1102

8/11/2019 Lecture Notes 1-SMES1102

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SMES1102Semester 2 2012/2013

Lecture notes 1:Vector


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Scalars and Vectors

Scalar: a quantity that has magnitude only.

Scalars may or may not have units associated

with them.

Vector: a quantity that has both magnitude

and direction


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Scalars and Vectors

length? force?

temperature?

velocity? voltage?

mass?

area?

displacement?

Scalar or vector?

Acceleration?

Momentum?


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Which of the following are vectors, scalars, or

neither?

a) 5.2 m/s left

b) downwards

c) 0.52 sd) 15 south of east

Scalars and Vectors


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Notation

A different notation is used for a different vector: A,

B, AB,

If the vector is handwritten, the variable symbol is

written with a right-pointing arrow over top: FEven if the vector itself is pointing left or down or northeast, the arrow on top of the

symbol still points right

If the vector appears in a textbook or other typesetdocument, the variable is usually bolded: F


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Graphical Representation

A vector is represented by a line segment (withmagnitude) and an assigned direction.

The magnitude or length of the vector is the length

of the segmentABand is denoted by .

Arrow is the direction from A to B

Note |AB| 0 (since lengths always 0)


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Graphical Representation

Consider the vectors in the diagram below

Vector B has the same magnitude as A, since it has the same length of 2 units on the grid.

Vector C has the same direction as A but a different magnitude, since it is 4 units long

A =B and A C

WARNING!!! The length of the arrow does not necessarily represent a length.

Similarly, E has the same direction as D but a smaller

magnitude, while F and G are in the opposite direction to D.

In fact, G is said to be the opposite or negative of D, because

it has the same magnitude, but the opposite direction


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Two vectors are equalif they have the same

magnitude and direction

A= BChas same direction as A, Bbut different magnitude

Dhas same magnitude as A, Bbut different direction

A B CD


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ahas same magnitude as a, but has opposite direction

So

| a| = |a|

The magnitude of a vector is ALWAYS positive


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Geometric Addition of Vectors

- Accomplished by connecting the vectors head to tail in sequence as

follow:

Following the commutative law:

The sum of two or more vectors is called the RESULTANT

This is defined as the method of adding any two vectors. It is called the TRIANGLE LAW


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Alternatively we can use the parallelogram law.


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+ =

But =so+=

This is called the PARALLELOGRAM LAW

Also note that since

+ = and = and =

We have +=

Hence +=+

so vector addition is commutative


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Using the following three vectors, find:

A B C

(a) A+ B

(b) A - B(c) A+ B+ C

(d) A+ B - C

Example 1:


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Associative Law for vectors is

(F1+ F2) + F3= F1+ (F2+ F3)

Proof for displacements:-


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Multiplication of a vector by a scalar

If we have (i) a scalar c > 0 and (ii) a vector A= 1, 2 , then we

define cAas follows.

cAis a vector.

Its magnitude is A =

Its direction is the same as A.

1, 2 = 1 2 + 2 2

= 12 + 2

2


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Similarly, if c< 0,

cA= 1, 2 is a vector with opposite direction from Aand magnitude

A .

1, 2 = 1, 2

A = A


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Example 2: Sketch u+ 2vusing vectors uand vin the figure


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A (a1, a2)

a1

a2

a

Position vector

For the position vector awith initial point at the origin and terminal point at the point

A (a1,a2), we denote the vector by

a= = 1, 2

a1and a2are the components of vector a

Magnitude of vector ais = 12 + 2

2 (Pythagorean Theorem)

If a vector ais represented in the plan with

initial point A (a1, a2) and terminal point O

(o1, o2), then a= o1a1, o2a1


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A (a1, a2)

B (b1, b2)

C (a1+ b1, a2+ b2)

O

1, 2 + 1, 2 = 1 + 1, 2 + 2

1, 2 - 1, 2 = 1 1, 2 2

Vector addition

Vector subtraction


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Vector Addition

a+ b= a+ b (commutativity)

a+ (b+ c) = (a+ b) + c (associativity)

a+ 0= a (zero vector)

a+ (-a) = 0 (additive inverse)

Multiplication by a Scalar

c(a+ b) = ca+ cb (distributive law)

(c+ d)a= ca+ du (distributive law)

(cd)a= c(da) = d(ca)

1a= a (multiplication by 1)

0a= 0 (multiplication by 0)

Length of a Vector

|ca| = |c| |a|

Properties of Vector


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Find the vector with:

(a) initial point at A (2,3) and terminal point at B (3,-1)

Example 3

= 3 2, 1 3 = 1 , 4

(b) initial point at B (3,-1) and terminal point at A (2,3)

= 2 3, 3 1 = 1 , 4


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Example 4: Express the vector with initial point P and terminal

point Q in component form


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Find a2band -3a+ 4bfor vectors

(i) a= 2, -5and b= -3, 1

(ii) a= 1, 7and b= 4, 1(iii) a= 5, -1and b= 3, 6(iv) a= 4, 9and b= 1, -2

Exercises


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Fundamental Cartesian Unit Vectors

Three special vectors play an important role and have a specialnotation. They are:-

A unit vector parallel to the X-axis, denoted by i

A unit vector parallel to the Y -axis, denoted by j

A unit vector parallel to the Z-axis, denoted by k

Any vector can be written in terms of i, j and k

Example : A= 2,3,4 = 2i + 3j+ 4k


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2D

u

Since OAis in the X-direction we can

write =p1i

Similarly AB=p2j

wherep1,p2are scalars.

u=p1 i+p2j


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3D

V

= OA+ AB+ BC=p1i+p2j+p3k


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Addition in terms of components

v1= a

1i+ b

1j+ c

1k

v2= a2i+ b2j+ c2k

v1+ v2= (a1+ a2)i+ (b1+ b2)j+ (c1+ c2)k


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Any vector can be broken down into components along the x and y axes.

Example: = 50m @ 30from the horizontal. Find its components.

= 4.3m i+ 2.5mj


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Consider the two vectors shown in the figure

below . Find thex andy components of the vectorsAand B

Vector problems


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Given three concurrent forces acting on a tent post. Find The magnitudeand angleof

the resultant force.

Vector problems


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1) A river flows at 3 km/h and a rower rows

at 6 km/h. What direction should the rower

take to go straight across a river?

Vector problems


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Products of Vectors

a) Scalar (dot) productthe result is a scalar

b) Vector productthe result is a vector

A.B= cos


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Scalar product

We define the scalar product (or dot product) of two vectors

Aand Bas

A

BA.B= cos

where is the angle between them and , are themagnitudes.

The dot product or scalar product yields a scalar bymultiplying two vectors. T


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Scalar product

(i) If Ais perpendicular to Bthen = 90and cos = 0,

so A.B= 0.

A

B

(ii) A.B= a2since = 0, and cos = 1.

(iii) i.i=j.j= k.k= 1 and i.j=j.k= k.i= 0

Special Cases

the result is alwas scalar!!!!!!


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Other Properties

Scalar product

1. For a scalar m

(ma).b= m(a.b) = a.(mb) = mabcos

2. (a.b)cis the product of a scalar a.bwith a vector c. Recallingthat mcis a vector in the direction of cwith magnitude mc we see

that (a.b)cis a vector in the direction of cwith magnitude

. .

3. a.(b+c) = a.b + a.cThe distributive law.


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Component form of the scalar product

Scalar product

Lecture Notes 1-SMES1102

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