Vector Algebra - An Introduction - Field Theory.


Most of the physical quantities are either scalar or vector quantities.


SCALAR QUANTITY:

- Scalar is a number that defines magnitude. Hence a scalar quantity is defined as a quantity that has magnitude only.

- A scalar quantity does not point to any direction i.e. a scalar quantity has no directional component.
For example when we say, the temperature of the room is 30o C, we don’t specify the direction.

- Hence examples of scalar quantities are mass, temperature, volume, speed etc.

- A scalar quantity is represented simply by a letter – A, B, T, V, S.


VECTOR QUANTITY:

- A Vector has both a magnitude and a direction. Hence a vector quantity is a quantity that has both magnitude and direction.

- Examples of vector quantities are force, displacement, velocity, etc.

- A vector quantity is represented by a letter with an arrow over it.


UNIT VECTORS (aA):

- When a simple vector is divided by its own magnitude, a new vector is created known as the unit vector
Mathematically,  aA = A / |A|

- A unit vector has a magnitude of one. Hence the name “unit vector”.

- A unit vector is always used to describe the direction of respective vector.
 Rearranging the terms, we have

- Hence any vector can be written as the product of its magnitude and its unit vector.

- Unit Vectors along the co-ordinate directions are referred to as the base vectors.
For example unit vectors along X, Y and Z directions are ax, ay and az respectively.


Position Vector / Radius Vector ( r ):

- A Position Vector (rQ)/ Radius vector defines the position of a point in space relative to the origin.
rQ = xax + yay +zaz

- If the coordinates of some point is given as x =1, y =2 and z =3, then the position vector is defined as
r = ax + 2ay +3az.

- Hence Position vector is another way to denote a point in space.

- Position vector for Cartesian system in general is written as
r = xax + yay +zaz

But we cannot say the position vector for cylindrical and spherical coordinate system to be

r = ρaρ + φaφ + zaz

r = rar +θaθ + φaφ
because θ and φ are not a unit of distance.

- Hence the correct position vector for cylindrical and spherical system is given as:

r = ρcosφax + ρsinφay + zaz

r = rsinθcosφax +rsinθsinφay + rcosθaz

- A position vector should always be expressed using Cartesian base vectors (ax, ay, az).

- Displacement Vector is the displacement or the shortest distance from one point to another.



SOLVED EXAMPLES / NUMERICALS:

Q.1. Given vectors A = ax + 3az and B = 5ax + 2ay -6az, determine

a) |A+B|
B) 5A – B
c) Component of A along ay.
d) Unit vector parallel to 3A + B.              SOLUTION/ANSWER



Q.2 Given Points P (1, -3, 5), Q (2, 4, 6) and R (0, 3, 8) find
a) Position vectors of P and R.
b) Distance vector rQR.
c) Distance between Q and R.                   SOLUTION/ANSWER



Q. 3 Find the unit vector along the line joining point (2, 4, 4) to point (-3, 2, 2) ?               SOLUTION/ANSWER


Q. 4 Given that A = 3ax + 5ay – 7az and B = ax – 2ay + az ; find
a) | 2B + 0.4A |
b) A.B - | B |2
c) A x B                                   SOLUTION/ANSWER



ALSO READ:

- Scalars and Vectors.

- Unit Vectors.

- Position and Distance Vectors.

- Vector Multiplication.

- Components Of a Vector.

- Line , Surface and Volume Intergral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).


Your suggestions and comments are welcome in this section. If you want to share something or if you have some stuff of your own, please do post them in the comments section.

Comments

  1. it would be nice to print this a one PDF document

    ReplyDelete
  2. It is incorrect to say that the position vector HAS to be expressed using Cartesian coordinates

    ReplyDelete
  3. Great that you people have been doing this. Don't stop, huh? If necessary, pass along the task to the younger generation...

    ReplyDelete

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