Vectors and Vector Arithmetic

ShatakirtiMT2011096

Vectors

Contents1 Scalars and Vectors 2

2 Vector Representaion 2

3 Vector Components 4

4 Vector Arithmetic 54.1 Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.1 Parallelogram Law of Addition . . . . . . . . . . . . . 74.1.2 Triangle Law of Addition . . . . . . . . . . . . . . . . . 74.1.3 Vector Subtraction . . . . . . . . . . . . . . . . . . . . 9

4.2 Properties of Vector Arithmetic . . . . . . . . . . . . . . . . . 104.3 Multiplication of Vectors . . . . . . . . . . . . . . . . . . . . . 10

4.3.1 Scalar Multiple of a Vector . . . . . . . . . . . . . . . . 104.3.2 Dot Product of Vectors . . . . . . . . . . . . . . . . . . 114.3.3 Cross Product of Vectors . . . . . . . . . . . . . . . . . 12

References 14

List of Figures1 Vector Representation . . . . . . . . . . . . . . . . . . . . . . 32 Vector Components . . . . . . . . . . . . . . . . . . . . . . . . 43 Net shift from A to C due to the two earthquakes on the town 54 Net shift from A to E due to the two earthquakes on the town 65 Commutative Property of Vector Addition . . . . . . . . . . . 66 Geometric Addition of Vectors . . . . . . . . . . . . . . . . . . 77 Parellogram Law of Addition . . . . . . . . . . . . . . . . . . . 78 Triangle Law of Addition of Vectors . . . . . . . . . . . . . . . 89 Three vectors are represented by three sides in sequence . . . 910 The resultant of three vectors represented by three sides is zero 911 Scalar Multiple of a Vector v . . . . . . . . . . . . . . . . . . 1012 Dot Product of Vectors a and b . . . . . . . . . . . . . . . . . 1113 Cross Product of Vectors a and b . . . . . . . . . . . . . . . . 12

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Vectors

1 Scalars and VectorsA scalar is a quantity that is completely specified by its magnitude and hasno direction. A scalar can be described either dimensionless, or in terms ofsome physical quantity. A vector is a quantity that specifies both a magni-tude and a direction. Such a quantity may be represented geometrically byan arrow of length proportional to its magnitude, pointing in the assigneddirection.

Example 1 : Take an example of jellybeans in a jar. If we had to knowhow many pounds of jellybeans we have, we could just give a value like 4pounds and we will have all the information we have asked for. Such quan-tities are called as scalar quantities. We didn′t have to say 4 pounds up or4 pounds right. We just needed to give the value. Such things have only asize, a magnitude, an amount. Other examples include time, volume, area,energy etc.- they don′t have a direction.Some quantities have direction too, and that is also important. Force is agood example of this. It has a direction. For example if we push an objectwith a certain force, say 500N, here, it makes a big difference in what direc-tion you are pushing it. Just saying giving 500N of force does not give you acomplete picture, you also need to know a direction as well.Things that are vectors are often called vector quantities. They have a mag-nitude (a bigness) AND a direction. For ex, force, velocity, acceleration etc.

2 Vector RepresentaionVector quantities are often represented by scaled vector diagrams. Vectordiagrams depict a vector by use of an arrow drawn to scale in a specificdirection. An example of a scaled vector diagram is shown in the diagrambelow.

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Vectors

Figure 1: Vector Representation

The following are the properties of the vector diagram :

1. A scale is clearly listed

2. A vector arrow (with arrowhead) is drawn in a specified direction. Thevector arrow has a head and a tail.

3. The magnitude and direction of the vector is clearly labeled. In thiscase, the diagram shows the magnitude is 20 m and the direction is (30degrees West of North).

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Vectors

3 Vector ComponentsAny vector directed in two dimensions can be thought of as having an in-fluence in two different directions. That is, it can be thought of as havingtwo parts. Each part of a two-dimensional vector is known as a component.The components of a vector depict the influence of that vector in a given di-rection. The combined influence of the two components is equivalent to theinfluence of the single two-dimensional vector. The single two-dimensionalvector could be replaced by the two components.

Figure 2: Vector Components

Example 2 : If a dog chain is stretched upward and rightward and pulledtight by his master, then the tension force in the chain has two components -an upward component and a rightward component. To the dog, the influenceof the chain on his neck is equivalent to the influence of two chains on hisbody - one pulling upward and the other pulling rightward. If the singlechain were replaced by two chains. with each chain having the magnitudeand direction of the components, then, the dog would not know the differencebecause the combined influence of the two components is equivalent to theinfluence of the single two-dimensional vector. Hence, any vector directed atan angle to the horizontal or the vertical can be thought of as having twoparts (or components). That is, any vector directed in two dimensions can bethought of as having two components. For example, if a chain pulls upwardat an angle on the collar of a dog, then there is a tension force directed in twodimensions. This tension force has two components: an upward componentand a rightward component.

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Vectors

4 Vector ArithmeticA variety of mathematical operations can be performed with and upon vec-tors. One such operation is the addition of vectors. Two vectors can beadded together to determine the result (or resultant).

Example 3 : Imagine an earthquake hits a town and all points in townmove 2 units east and 1 units north. That means every point in the town hasshifted by this same amount. In the figure, point A in the town has shifted2 units east to B and 1 units north to C. hence the point A in the town hasmoved to point C after the earthquake. Similarly all the points in the townhave moved the same as shown in the fig below.

Figure 3: Net shift from A to C due to the two earthquakes on the town

We have 2 displacement vectors with magnitude and direction of 2 units,East and 1 unit, north. These can be added together to produce a resultantvector that is directed both East and North. When the two vectors are addedhead-to-tail, the resultant is the hypotenuse of a right angle triangle. Thesides of the right triangle will have lengths of 2 units and 1 unit.

Example 4 : Now, suppose in the above example, after the town hasbeen hit by the earthquake, every point in the town has moved 2 units eastand one point north, and later the town has been hit again by another quakeand it moves 3 units to the east and 4 units to the south. Let′s representthe first quake with the vector v =<2, 1> and the second quake with thevector w=<3, -4> Now we have that V moves everything 2 units right andW moves everything 3 units right. And thus the net shift is 5 units to theright. Then,v moves everything 1 unit upward and w moves everything 4units down, thus the resultant net shift is 1-4=-3 units. i.e., 3 units down.Thus v + w = <2, -3>

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Vectors

Figure 4: Net shift from A to E due to the two earthquakes on the town

Algebraically we can say that the sum of two vectors is simply the ad-dition of the x components and the y components. And geometrically, it isjust one vector followed by another (in this ex. It is v followed by w)

We can even see that vector addition commute i.e. if we hit the townwith the earthquake w first and the earthquake v second, we would stillget the resultant net shift as the same as shown in the fig. below we get aparallelogram ACEF.

Figure 5: Commutative Property of Vector Addition

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Vectors

4.1 Vector Addition

Figure 6: Geometric Addition of Vectors

4.1.1 Parallelogram Law of Addition

The parallelogram law states that, if two adjacent sides of a parallelogramrepresents two given vectors in magnitude and direction, then the diagonalstarting from the intersection of two vectors represent their sum. The exam-ple of law of parallelogram of vector addition is given in following picture:

Figure 7: Parellogram Law of Addition

4.1.2 Triangle Law of Addition

The law of triangle of vector addition states that if two vectors are representedby two sides of a triangle in sequence, then third closing side of the triangle,in the opposite direction of the sequence, represents the sum (or resultant)of the two vectors in both magnitude and direction.

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Vectors

Here, the term ”sequence” means that the vectors are placed such thattail of a vector begins at the arrow head of the vector placed before it. Forexample: Let there be two vectors A and B and the angle between them isθ as shown in the picture below

Figure 8: Triangle Law of Addition of Vectors

Then, To find their sum(a + b) first of all we reposition the two vectorssuch that the head of vector a exactly coincides with the tail of vector b orvice versa and then draw a vector c as shown in the figure, the newly drawnvector c represents the sum of vectors a and b.

If three vectors are represented by three sides of a triangle in sequence,then resultant vector is zero. In order to prove this, let us consider any twovectors in sequence like AB and BC as shown in the figure. According totriangle law of vector addition, the resultant vector is represented by thethird closing side in the opposite direction. It means that : AB+BC=AC

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Vectors

Figure 9: Three vectors are represented by three sides in sequence

Adding vector CA on either sides of the equation,AB+BC+CA=AC+CA

The right hand side of the equation is vector sum of two equal and oppo-site vectors, which evaluates to zero. Hence, AB+BC+CA=0

Figure 10: The resultant of three vectors represented by three sides is zero

4.1.3 Vector Subtraction

To define the subtraction of vectors first we need to define the negative vectorof a vector. The negative vector of vector A is denoted by vector -A and isa vector with the same magnitude as of vector A, but with exactly oppositedirection.

Adding -b has the same effect as subtracting b , so we use the followingformula to subtract a vector from another: a - b = a + (-b)

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Vectors

4.2 Properties of Vector ArithmeticIf u, v and w are vectors in 2-space or 3-space and c and k are scalars then,

1. u + v = v + u

2. u + (v + w) = (u + v) + w

3. u + 0 = 0 +u = u

4. u - u = u + (-u) = 0

5. 1u = u

6. (ck)u = c(ku) = k(cu)

7. (c + k)u = cu + ku

8. c(u + v) = cu + cv

4.3 Multiplication of Vectors4.3.1 Scalar Multiple of a Vector

Suppose that v is a vector and c is a non-zero scalar (i.e. c is a number) thenthe scalar multiple, cv, is the vector whose length is times the length of vand is in the direction of v if c is positive and in the opposite direction of vis c is negative.

Figure 11: Scalar Multiple of a Vector v

Note that we can see from this that scalar multiples are parallel. In factit can be shown that if v and w are two parallel vectors then there is a

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non-zero scalar c such that , or in other words the two vectors will be scalarmultiples of each other.

It can also be shown that if v is a vector in either 2-space or 3-space thenthe scalar multiple can be computed as follows,cv=(cv1,cv2 ) OR cv=(cv1,cv2,cv3 )

4.3.2 Dot Product of Vectors

The dot product of two vectors shows the projection of one vector on theother. For ex. if we have 2 vectors a and b, the dot product of the twovectors is given by,a.b = |a| |b| cosθwhere θ is the angle between the two vectors.

Figure 12: Dot Product of Vectors a and b

From the above formula, we can deduce some properties of the dot prod-uct of two vectors :

1. If both a and b have length one (i.e., they are unit vectors), their dotproduct simply gives the cosine of the angle between them.

2. If only b is a unit vector, then the dot product a.b gives |a| cosθ, i.e.,the magnitude of the projection of a in the direction of b, with a minussign if the direction is opposite. This is called the scalar projection ofa onto b, or scalar component of a in the direction of b.

3. If neither a nor b is a unit vector, then the magnitude of the projectionof b in the direction of a is a. b

|b| , as the unit vector in the direction ofb is b

|b|

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Vectors

Applications of dot productIn general the dot product is used whenever we need to project a vector ontoanother vector. Some concrete examples where the dot product is used aspart of the solution are :

1. Calculate the distance of a point to a line

2. Calculate the distance of a point to a plane

3. Calculate the projection of a point on a plane

4. Find the component of one vector in the direction of another

4.3.3 Cross Product of Vectors

The cross product of vectors a and b is a vector perpendicular to both aand b and has a magnitude equal to the area of the parallelogram generatedfrom a and b. The direction of the cross product is given by the right-handrule. The cross product is denoted by a ”x” between the vectors.

(a) Cross product representation (b) Right hand Thumb rule

Figure 13: Cross Product of Vectors a and b

The cross product is defined by the formula :a x b = ab sinθ nwhere θ is the measure of the smaller angle between a and b (0◦≤θ≤180◦),a and b are the magnitudes of vectors a and b (i.e., a = |a| and b = |b|),and n is a unit vector perpendicular to the plane containing a and b in thedirection given by the right-hand rule as illustrated above.

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Vectors

Applications of cross productSome examples where the cross product is used as part of the solution are :

1. Calculate the Area of parallelogram

2. Calculate the Volume of parallelopiped

3. Cross product is vastly used in physics in different applications likefinding the Torque or the Momentum of a Force.

4. It shows the vectors′ relationship in the plane in which they lie. Thisis very important for studying the surfaces which can be seen as planesin small scales. This is used for the study of flux lines for electric andmagnetic fields.

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References[1] en.wikipedia.org/wiki/Vector

[2] en.wikipedia.org/wiki/Dot_product

[3] en.wikipedia.org/wiki/Cross_product

[4] http://www.physicsclassroom.com/class/1dkin/u1l1b.cfm

[5] http://tutorial.math.lamar.edu/Classes/LinAlg/Vectors.aspx

[6] http://knol.google.com/k/dot-product-cross-product-in-3d#

[7] http://mathworld.wolfram.com/VectorAddition.html

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