Scalars and Vectors

Mathematics is the language of Physics.

Scalars and Vectors
Scalar Quantities can be denoted by a number and a unit, i.e. Magnitude. Vector Quantities are denoted by magnitude and direction. Tensor Quantities

Properties of Vectors
Equal Vectors

Vectors having same direction and magnitude.

Parallel Vectors

Vectors having same direction.

Anti-Parallel Vectors

Vectors having opposite directions.

Position Vector

Vector drawn from the origin to a point.

Zero Vector

Vector of magnitude zero.

Free Vector

Vector which can be moved through space, without changing it's direction.

Triangle Law
If two vectors exist such that their directions are taken in order, then their resultant is equal to the third of the triangle formed ; & the direction is from head of first to tail of second.

Parallelogram Law
The parallelogram law of Vectors states that if two vectors originate or meet at a common point, then their resultant is given by the diagonal of parallelogram formed.

Scalar Multiplication
Scalar multiplication refers to the multiplication of a vector with a number having no direction.

In Scalar Multiplication, the number is multiplied to all components of the vector. Thus, scalar multiplication does not change the direction of the vector , it only affects the magnitude.

Resolution of Vectors
for co-ordinate system, we know that :  x = rcosθ   y = rsinθ

for 3-d coordinate system, we resolve using direction cosines : r = acosα + acosβ + acosγ

Scalar Product
a.b = abcosθ

Scalar Product gives us a Scalar result.

Vector Product
axb = absinθ

Vector product gives us a vector result.

Unit Vector
A unit vector has unit magnitude and any direction. If we divide a vector by it's magnitude, we get an unit vector.

Differential Calculus
Rate Measurer : Using graphs in physics, we can find the rate of change of quantity on Y-Axis w.r.t. change in quantity on x-Axis by differentiating the function and applying the values at that point. or by finding the slope of tangent at that point.

Maxima & Minima : We can interpret data from the graph by finding it's maximum & minimum value by second order derivative test.

Tips & Tricks
1) If in any question, based on any chapter , direction is being asked , then always work with vectors (xi + yj + zk)