VECTORS
How to Locate Points in Space?
Positions in space are designated relative to coordinate systems. The Cartesian coordinate system is a particularly convenient coordinate system in which position is designated by distances (x, y, z) along three perpendicular axes that intersect at a point called the origin
In a polar coordinate system positions in a plane are designated by a length r from the origin, and an angle q usually measured from the positive x - axis. From simple trigonometry we see that the relationships between the polar coordinates and the Cartesian coordinates are
x=rcosθ
y=rsinθ
A frame of reference is another name for the particular coordinate system with respect to which we are making observations of physical phenomena.
A scalar quantity requires only a number for its complete description. Mass, volume, density, pressure and temperature are all examples of scalar quantities. The mathematics of scalar quantities is the ordinary algebra of numbers.
Vector quantities require both magnitude and direction for its complete description. Velocity, acceleration, force and momentum are the examples of vector quantities. A vector can be represented graphically by a directed line segment. The length of the line segment represents the vector's magnitude and its angle with respect to some coordinate system specifies its direction.
We will represent vectors by bold face type letters, with an arrow over the letter such as a⃗ . When written by hand the same representation may be used.
The magnitude of a vector will be represented by italic type letters such as a, b, c etc.
If two vectors have the same direction, they are parallel. If they have the same magnitude and the same direction, they are equal, no matter where they are located in space.
If two vectors have the same direction, they are parallel. If they have the same magnitude and the same direction, they are 
equal, no matter where they are located in space.
The negative of a vector is defined as a vector having the same magnitude as the original vector but the opposite direction.
When two vectors and have opposite directions, whether their magnitudes are the same or not, we say that they are antiparallel.
Vector Algebra
Addition of Vectors
(i) Geometrical Method
Two vectors a⃗ and b⃗ may be added geometrically by drawing them to a common scale and placing then head to tail. The vector connecting the tail of the first to the head of the second is the sum vector c⃗ .
Vector addition is commutative and obeys the associative law.
(ii) Analytical Method (Parallelogram law of vector addition)
If the two vectors a⃗ and b⃗ are given such that the angle between them is θ. The magnitude of the resultant vector c⃗ of their vector addition is given by
c=a2+b2+2abcosθ−−−−−−−−−−−−−−−√
and its direction is given by angle Ï• with
vector a⃗
tanϕ=bsinθa+bcosθ
It is a common error to conclude that if c⃗ =a⃗ +b⃗ the magnitude of c⃗ should be just equal to the magnitude of a⃗ plus the magnitude of b⃗ In general, the conclusion is wrong; one can see that c < a + b. The magnitude of the vector sum a⃗ +b⃗ depends on the magnitudes of a⃗ and of b⃗ and on the angle between a⃗ and b⃗ Only in the special case in which a⃗ and b⃗ are parallel; the magnitude of c⃗ =a⃗ +b⃗ equal to the sum of the magnitudes of a⃗ and b⃗ By contrast, when the vectors are anti-parallel the magnitude of c⃗ equals the difference of the magnitudes of a⃗ and b⃗ .
Subtraction of Vectors
To subtract b⃗ from a⃗ , reverse the direction of b⃗ to get -b⃗ ; then add -b⃗ to a⃗
Resolution of a Vector
The (scalar) components, ax and ay of any two dimensional vector a⃗ are found by dropping perpendicular lines from the ends of a⃗ onto
the coordinate axes. The components are given by
ax=acosθ
and ay=asinθ
where θ is the angle from the positive direction of the x axis to the direction of a⃗ The algebraic sign of a component indicates its direction along the associated axis.
Given the components, we can reconstruct the vector from
a=a2x+a2y−−−−−−√
and tanθ=ayax
Application 1
Find the components of the vectors a⃗ and b⃗ as shown in the figure, if a = 2 units and b = 3 units.
Solution
ax=acosθ=2cos30∘=1.73 units
ay=asinθ=2sin30∘=1 units
bx=bcosθ=2cos45∘=2.12 units
by=−bsinθ=−3sin45∘=−2.12 units
Application 2
Find the magnitude and direction of the resultant vector c⃗ =a⃗ +b⃗ where a⃗ and b⃗ are vectors shown in figure.
Solution
cx=ax+bx=1.73+2.12=3.85 units
cy=ay+by=1−2.12=−1.12 units
c=c2x+c2y−−−−−−√=(3.85)2+(−1.12)2−−−−−−−−−−−−−−−√ = 4 units
tanθ=cycx=−1.123.85=−0.29 units
θ=tan−1(−0.29)
Note
There is one slight complication in using equation tanθ=ayax to find θ Suppose ax=2 m and ay=−2 m; then tanθ=−1 But there are two angles having tangents of -1, namely 135° and 315° (or -45°). In general, any two angles that differ by 180° have the same tangent. To decide which is correct, we have to look at the individual components. Because ax is positive and ay is negative, the angle must be in the fourth quadrant; thus θ= 315° (or -45°) is the correct value. But if instead we have ax = -2 m and ay = 2 m, then the correct angle is 135°. Similarly, when ax and ay are both negative, the tangent is positive, but the angle is in the third quadrant. One should always draw a sketch to check which of the two possibilities is the correct one.
Unit Vectors
A unit vector is a vector that has a magnitude of 1, with no units. Its only purpose is to point, that is, to describe a direction in space. Unit vectors provide a convenient notation for many expressions involving components of vectors
In an x-y coordinate system we can define a unit vector i^ that points in the direction of the positive x - axis and a unit vector j^ that points in the direction of the positive y-axis. Then we can express the relationship between component vectors and components as
a⃗ x=axi^
The vector a⃗ can be written in terms of its components as
a⃗ x=axi^+ayj^
When two vectors a⃗ and b⃗ are represented in terms of their components, we can express the vector sum c⃗ using unit vector as follows
a⃗ =axi^+ayj^
b⃗ =bxi^+byj^
c⃗ =(axi^+ayj^)+(bxi^+byj^)= c⃗ =(ax+bx)i^+(ay+by)j^
In three dimensional coordinate system a vector a⃗ may be expressed as
a⃗ =axi^+ayj^+azk^
where k^ is the unit vector along the z-axis.
Application 3
If a⃗ =2i^+j^ and b⃗ =4i^+7j^
(a) Find the components of c⃗ =a⃗ +b⃗
(b) Find the magnitude of c⃗ and its angle with x-axis.
Solution
(a) c⃗ =a⃗ +b⃗ =(2i^+j^)+(4i^+7j^)
c⃗ =(2+4)i^+(1+7)j^
Thus cx=6; cy=8
(b) c=c2x+c2y−−−−−−√=(6)2+(8)2−−−−−−−−−√=100
tanθ=cycx=86=43
θ=tan−1(43)=53∘
Multiplication of Vectors
(i) Multiplication of a vector by a scalar
The product of a scalar b and a vector v⃗ is a new vector whose magnitude is b v⃗ and whose direction is the same as that of v⃗ if b is positive, and opposite to that of v⃗ if b is negative. To divide v⃗ by b, multiply v⃗ by (1/b).
(ii) Multiplication of a vector by another vector
(a) Scalar or Dot Product
The scalar (or dot) product of two vectors a⃗ and b⃗ is written a⃗ .b⃗ and is the scalar quantity c given by
c=a⃗ .b⃗ =abcosθ
In which θ is the angle between the directions of a⃗ and b⃗ . The scalar product may be positive, zero, or negative, depending on the value of θ. A scalar product is the product of magnitude of one vector and the component of the second vector along the direction of the first vector.
Some properties of Dot product
1) It is commutative, i.e. a⃗ .b⃗ = b⃗ .a⃗
2) It is distributive over addition and subtraction i.e. a⃗ .(b⃗ ±c⃗ )=a⃗ .b⃗ ±a⃗ .c⃗
3) If a⃗ and b⃗ are perpendicular a⃗ .b⃗ = 0.
4) If a⃗ and b⃗ are parallel then a⃗ .b⃗ =a.b
5) Square of a vector is defined as a⃗ .a⃗ = a2 (scalar) and not a⃗ ×a⃗
i^.i^=j^.j^=k^.k^=(1)(1)cos0∘=1
i^.j^=j^.k^=k^.i^=(1)(1)cos90∘=0
6) In unit-vector notation we have,
a⃗ .b⃗ =(axi^+ayj^+azk^).(bxi^+byj^+bzk^)
Thus, a⃗ .b⃗ =axbx+ayby+azbz
Application 4
If a⃗ =2i^+3j^; b⃗ =4i^+2j^ Find c=a⃗ .b⃗
Solution
c⃗ =a⃗ .b⃗ =(2i^+3j^)+(4i^+2j^)=(2)(4)+(3)(2)=14
(b) Vector or Cross product
The vector (or cross) product of two vectors a⃗ and b⃗ is written as a⃗ ×b⃗ and is a vector c⃗ whose magnitude c is given by
c=absinθ
In which θ is the smaller of the angles between the direction of a⃗ and b⃗ . the direction of c⃗ is perpendicular to the plane defined by a⃗ and b⃗ and is given by right hand rule.
c⃗ =a⃗ ×b⃗
|c⃗ |=a⃗ ×b⃗ = absinθ
and a⃗ ×b⃗ =|a⃗ |∣∣b⃗ ∣∣sinθn^
where n^ is the unit vector.
The vector c⃗ is directed perpendicular to the plane formed by a⃗ and b⃗ The direction of vector c⃗ may be obtained by using the Right Hand Thumb Rule. Stretch all the fingers and thumb of your right hand such that they are perpendicular to each other. Now align your hand such that its plane is perpendicular to the plane formed by vectors a⃗ and b⃗ . Align the stretched fingers along the direction of the vector written first in order i.e., a⃗ in this case . Curl the fingers of your hand towards the second vector through the smaller angle. Then, the direction of the thumb gives the direction of the cross product.
Properties of Cross Product
1) It is not commutative a⃗ ×b⃗ ≠b⃗ ×a⃗ . In fact a⃗ ×b⃗ =−b⃗ ×a⃗
2) It is distributive over addition and subtraction
a⃗ ×(b⃗ ±c⃗ )=a⃗ ×b⃗ ±a⃗ ×c⃗
3) If a⃗ ∥b⃗ , then a⃗ ×b⃗ =0 and if a⃗ ⊥b⃗ , ∣∣a⃗ ×b⃗ ∣∣=ab
4) If i^, j^, k^ be the unit vectors along in the positive directions of x, y, and z axes then
i^×i^=j^×j^=k^×k^=0
i^×j^=k^; j^×k^=i^; k^×i^=j^ (maintain cyclic order) i^×j^=−j^×i^
5) The cross product may also be expressed by the determinant
a⃗ ×b⃗ =∣∣∣∣∣i^axbxj^aybyk^azbz∣∣∣∣∣
where a⃗ =axi^+ayj^+azk^
and b⃗ =bxi^+byj^+bzk^
Application 5
If a⃗ =2i^+3j^; b⃗ =4i^+2j^ Find d⃗ =a⃗ ×b⃗
Solution
d⃗ =(2i^+3j^)×(4i^+2j^)
d⃗ =(2)(4)(i^×i^)+(2)(2)(i^×j^)+(3)(4)(j^×i^)+(3)(2)(j^×j^)
Since i^×i^=0 ; j^×j^=0
i^×j^=k^ ; j^×i^=−k^
d⃗ =4k^−12k^=−8k^
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