Essay on Vectors and Vector Algebra

1. Vectors and Vector Arithmetic
Shatakirti
MT2011096

2. Vectors
Contents
1 Scalars and Vectors 2
2 Vector Representaion 2
3 Vector Components 4
4 Vector Arithmetic 5
4.1 Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.1.1 Parallelogram Law of Addition . . . . . . . . . . . . . 7
4.1.2 Triangle Law of Addition . . . . . . . . . . . . . . . . . 7
4.1.3 Vector Subtraction . . . . . . . . . . . . . . . . . . . . 9
4.2 Properties of Vector Arithmetic . . . . . . . . . . . . . . . . . 10
4.3 Multiplication of Vectors . . . . . . . . . . . . . . . . . . . . . 10
4.3.1 Scalar Multiple of a Vector . . . . . . . . . . . . . . . . 10
4.3.2 Dot Product of Vectors . . . . . . . . . . . . . . . . . . 11
4.3.3 Cross Product of Vectors . . . . . . . . . . . . . . . . . 12
References 14
List of Figures
1 Vector Representation . . . . . . . . . . . . . . . . . . . . . . 3
2 Vector Components . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Net shift from A to C due to the two earthquakes on the town 5
4 Net shift from A to E due to the two earthquakes on the town 6
5 Commutative Property of Vector Addition . . . . . . . . . . . 6
6 Geometric Addition of Vectors . . . . . . . . . . . . . . . . . . 7
7 Parellogram Law of Addition . . . . . . . . . . . . . . . . . . . 7
8 Triangle Law of Addition of Vectors . . . . . . . . . . . . . . . 8
9 Three vectors are represented by three sides in sequence . . . 9
10 The resultant of three vectors represented by three sides is zero 9
11 Scalar Multiple of a Vector v . . . . . . . . . . . . . . . . . . 10
12 Dot Product of Vectors a and b . . . . . . . . . . . . . . . . . 11
13 Cross Product of Vectors a and b . . . . . . . . . . . . . . . . 12
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3. Vectors
1 Scalars and Vectors
A scalar is a quantity that is completely specified by its magnitude and has
no direction. A scalar can be described either dimensionless, or in terms of
some physical quantity. A vector is a quantity that specifies both a magni-
tude and a direction. Such a quantity may be represented geometrically by
an arrow of length proportional to its magnitude, pointing in the assigned
direction.
Example 1 : Take an example of jellybeans in a jar. If we had to know
how many pounds of jellybeans we have, we could just give a value like 4
pounds 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 or
4 pounds right. We just needed to give the value. Such things have only a
size, 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 a
good example of this. It has a direction. For example if we push an object
with 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 a
complete 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 Representaion
Vector quantities are often represented by scaled vector diagrams. Vector
diagrams depict a vector by use of an arrow drawn to scale in a specific
direction. An example of a scaled vector diagram is shown in the diagram
below.
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4. 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. The
vector arrow has a head and a tail.
3. The magnitude and direction of the vector is clearly labeled. In this
case, the diagram shows the magnitude is 20 m and the direction is (30
degrees West of North).
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5. Vectors
3 Vector Components
Any 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 having
two 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 the
influence of the single two-dimensional vector. The single two-dimensional
vector could be replaced by the two components.
Figure 2: Vector Components
Example 2 : If a dog chain is stretched upward and rightward and pulled
tight by his master, then the tension force in the chain has two components -
an upward component and a rightward component. To the dog, the influence
of the chain on his neck is equivalent to the influence of two chains on his
body - one pulling upward and the other pulling rightward. If the single
chain were replaced by two chains. with each chain having the magnitude
and direction of the components, then, the dog would not know the difference
because the combined influence of the two components is equivalent to the
influence of the single two-dimensional vector. Hence, any vector directed at
an angle to the horizontal or the vertical can be thought of as having two
parts (or components). That is, any vector directed in two dimensions can be
thought of as having two components. For example, if a chain pulls upward
at an angle on the collar of a dog, then there is a tension force directed in two
dimensions. This tension force has two components: an upward component
and a rightward component.
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6. Vectors
4 Vector Arithmetic
A variety of mathematical operations can be performed with and upon vec-
tors. One such operation is the addition of vectors. Two vectors can be
added together to determine the result (or resultant).
Example 3 : Imagine an earthquake hits a town and all points in town
move 2 units east and 1 units north. That means every point in the town has
shifted by this same amount. In the figure, point A in the town has shifted
2 units east to B and 1 units north to C. hence the point A in the town has
moved to point C after the earthquake. Similarly all the points in the town
have 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 resultant
vector that is directed both East and North. When the two vectors are added
head-to-tail, the resultant is the hypotenuse of a right angle triangle. The
sides of the right triangle will have lengths of 2 units and 1 unit.
Example 4 : Now, suppose in the above example, after the town has
been hit by the earthquake, every point in the town has moved 2 units east
and one point north, and later the town has been hit again by another quake
and it moves 3 units to the east and 4 units to the south. Let s represent
the first quake with the vector v =<2, 1> and the second quake with the
vector w=<3, -4> Now we have that V moves everything 2 units right and
W moves everything 3 units right. And thus the net shift is 5 units to the
right. Then,v moves everything 1 unit upward and w moves everything 4
units 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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7. 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 is
just 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 town
with the earthquake w first and the earthquake v second, we would still
get the resultant net shift as the same as shown in the fig. below we get a
parallelogram ACEF.
Figure 5: Commutative Property of Vector Addition
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8. 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 parallelogram
represents two given vectors in magnitude and direction, then the diagonal
starting 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 represented
by 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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9. Vectors
Here, the term ”sequence” means that the vectors are placed such that
tail of a vector begins at the arrow head of the vector placed before it. For
example: Let there be two vectors A and B and the angle between them is
θ 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 vectors
such that the head of vector a exactly coincides with the tail of vector b or
vice versa and then draw a vector c as shown in the figure, the newly drawn
vector 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 two
vectors in sequence like AB and BC as shown in the figure. According to
triangle law of vector addition, the resultant vector is represented by the
third closing side in the opposite direction. It means that : AB+BC=AC
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10. 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 vector
of a vector. The negative vector of vector A is denoted by vector -A and is
a vector with the same magnitude as of vector A, but with exactly opposite
direction.
Adding -b has the same effect as subtracting b , so we use the following
formula to subtract a vector from another: a - b = a + (-b)
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11. Vectors
4.2 Properties of Vector Arithmetic
If 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 Vectors
4.3.1 Scalar Multiple of a Vector
Suppose that v is a vector and c is a non-zero scalar (i.e. c is a number) then
the scalar multiple, cv, is the vector whose length is times the length of v
and is in the direction of v if c is positive and in the opposite direction of v
is c is negative.
Figure 11: Scalar Multiple of a Vector v
Note that we can see from this that scalar multiples are parallel. In fact
it can be shown that if v and w are two parallel vectors then there is a
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12. Vectors
non-zero scalar c such that , or in other words the two vectors will be scalar
multiples of each other.
It can also be shown that if v is a vector in either 2-space or 3-space then
the 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 the
other. For ex. if we have 2 vectors a and b, the dot product of the two
vectors is given by,
a.b = |a| |b| cosθ
where 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 dot
product 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 minus
sign if the direction is opposite. This is called the scalar projection of
a 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 projection
b
of b in the direction of a is a. |b| , as the unit vector in the direction of
b
b is |b|
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13. Vectors
Applications of dot product
In general the dot product is used whenever we need to project a vector onto
another vector. Some concrete examples where the dot product is used as
part 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 a
and b and has a magnitude equal to the area of the parallelogram generated
from a and b. The direction of the cross product is given by the right-hand
rule. 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 n
where 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 the
direction given by the right-hand rule as illustrated above.
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14. Vectors
Applications of cross product
Some 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 like
finding the Torque or the Momentum of a Force.
4. It shows the vectors relationship in the plane in which they lie. This
is very important for studying the surfaces which can be seen as planes
in small scales. This is used for the study of flux lines for electric and
magnetic fields.
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15. Vectors
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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