What is a Vector?

Understanding Vectors in a Plane

A vector is a quantity that has both magnitude and direction. In the Cartesian plane, vectors are often represented as directed line segments.

Key characteristics of vectors include:

• Magnitude: The length of the vector, representing the size of the quantity.
• Direction: The angle or orientation of the vector in the plane.
• Zero Vector: A vector with zero magnitude and no specific direction, represented as $\mathbf{0}$.
• Parallel Vectors: Vectors that have the same or opposite direction.

 

Vector Notation

Vectors are usually denoted by bold letters or with an arrow above, such as $\mathbf{a}$ or $\vec{a}$.

In component form, a vector $\vec{a}$ in a plane can be written as:

$$\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$$

where $a_1$ and $a_2$ are the components of the vector along the x-axis and y-axis, respectively.

 

Examples of Vectors

Vectors can represent various physical quantities, such as:

• Displacement: The change in position of an object.
• Velocity: The speed of an object in a specific direction.
• Force: A push or pull acting on an object.