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 30^o 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 (a_A):

- When a simple vector is divided by its own magnitude, a new vector is created known as the unit vector
Mathematically,  a_A = 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 a_x, a_y and a_z respectively.


Position Vector / Radius Vector ( r ):

- A Position Vector (r_Q)/ Radius vector defines the position of a point in space relative to the origin.
r_Q = xa_x + ya_y +za_z

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

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

- Position vector for Cartesian system in general is written as
r = xa_x + ya_y +za_z

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

r = ρa_ρ + φa_φ + za_z

r = ra_r +θa_θ + φa_φ
because θ and φ are not a unit of distance.

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

r = ρcosφa_x + ρsinφa_y + za_z

r = rsinθcosφa_x +rsinθsinφa_y + rcosθa_z

- A position vector should always be expressed using Cartesian base vectors (a_x, a_y, a_z).

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



SOLVED EXAMPLES / NUMERICALS:

Q.1. Given vectors A = a_x + 3a_z and B = 5a_x + 2a_y -6a_z, determine

a) |A+B|
B) 5A – B
c) Component of A along a_y.
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 r_QR.
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 = 3a_x + 5a_y – 7a_z and B = a_x – 2a_y + a_z ; 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).


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