# Scalars and Vectors

**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**

Tensor Quantities are those quantities whose magnitude changes based upon the direction in which you measure it. (e.g. Moment of Inertia)

**Typ****es 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 its 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.

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 its magnitude, we get an unit vector.