Definition of a Vector

1. Formal Definition

A vector is a quantity that has both:

• magnitude (size or numerical value)
• direction (orientation in space)

This distinguishes it from a scalar, which has magnitude only.

2. Mathematical Representation

A vector is commonly written as:

\( \vec{A} \)

or in component form:

\( \vec{A}=(A_{x},A_{y},A_{z}) \)

or using standard basis vectors:

\( \vec{A}=A_x\hat{\imath} + A_y\hat{\jmath} + A_z\hat{k} \)

This representation encodes both:

• how much (magnitude)
• which way (direction)

3. Geometric Interpretation

Geometrically, a vector is represented as an arrow:

• length → magnitude
• direction of arrow → direction of the vector

The vector is independent of its position in space (for free vectors), meaning it can be shifted without changing its meaning.

4. Examples in Mechanics

Common vector quantities include:

• displacement
• velocity
• acceleration
• force

Each requires both magnitude and direction to fully describe the physical situation.

5. Key Idea

A vector is not just a number—it is a directed quantity that describes physical behavior in space, making it essential for analyzing motion and forces in mechanics.