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# What Are Vectors?

The study of vectors is useful in engineering mathematics, physics, civil engineering, research fields etc.

Vector calculus was developed by Josiah Willard Gibbs around the 19th century.

## Definition-

When a quantity has magnitude as well as direction (both) is called a vector.

Note- A physical quantity having magnitude only, is known as scalar.

For example- Acceleration, force, velocity, momentum, magnetic induction etc. are the vector quantities.

and density, speed, energy, mass, volume, temperature etc. are the examples of scalar quantities.

## Characteristics of vectors-

• A vector quantity has both direction and magnitude.
• We use an arrow sign on the head of the symbol we use for vector quantity.
• Vectors do not follow the algebra rules when they are added or subtracted.

Suppose we have two vectors and then we can add these two by joining them head to tail as below-

is called the sum of vectors and .

OR

Note-

• Addition of vectors satisfies commutative and associative law.
• The arrows show directions of the vector.

## Subtraction of vectors-

We can subtract one vector from another by reversing the direction of vector that we need to subtract and adding them-

See below-

## Unit vector-

Suppose a vector is then the unit vector is defined as-

## Product of two vectors-

There are two types of product of two vectors, the first one is scalar product or dot product and other is vector product or cross product.

The scalar or dot product is defined as-

Scalar product-

Suppose there are two vectors and then their scalar product is-

Note-

The dot product is commutative –

.=.

Vector product –

The vector product or cross product is defined as-

here is the unit vector in the direction of