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Vector Types

Vector Magnitude and Direction

Every vector has two main components: magnitude (length) and direction.

Consider the example vector $\overrightarrow{CD}$ below.

Vector

CD

Vector TypesLearn zero, negative, and equivalent vectors, including their magnitude, direction, properties, and example exercises.

On this page

Vector Magnitude and Direction
Negative Vector (Opposite Vector)
Zero Vector
Equivalent Vectors (Equal Vectors)
Exercise

Comments

Vector

CD

with a length of

4 \text{ cm}

and a direction of

45^\circ

relative to the horizontal line.

The magnitude of vector $\overrightarrow{CD}$ is $4 \text{ cm}$ . This indicates the size or value of the vector. The magnitude of a vector $\vec{v}$ is usually denoted as $|\vec{v}|$ . So, $|\overrightarrow{CD}| = 4 \text{ cm}$ .
The direction of vector $\overrightarrow{CD}$ is $45^\circ$ relative to the horizontal line. This direction is crucial and distinguishes vectors from scalar quantities (which only have magnitude). Direction can be expressed using angles, compass points (like Northeast), or other references.

Negative Vector (Opposite Vector)

A negative vector or opposite vector is a vector that has the same magnitude but opposite direction to the original vector.

Imagine Andi walks $100 \text{ m}$ in a direction of $30^\circ$ (let's call this displacement vector $\vec{A}$ ). Then, Andi returns to the starting position. This second displacement is the opposite vector of $\vec{A}$ , which we write as $-\vec{A}$ .

Vector and Opposite Vector

Vector

A

and vector

-A

have the same magnitude but opposite directions.

Vectors $\vec{A}$ and $-\vec{A}$ have the same magnitude ( $|\vec{A}| = |-\vec{A}|$ ).
The direction of vector $-\vec{A}$ is exactly opposite to the direction of vector $\vec{A}$ . If points in one direction, points in the opposite direction ( difference).

Zero Vector

The zero vector is a special vector because it has zero magnitude. Due to its zero length, this vector does not have a specific direction.

The zero vector can be visualized as a single point, where the initial point and terminal point coincide. The zero vector is usually denoted by $\vec{0}$ .

Example:

If Andi walks $100 \text{ m}$ east, then walks back $100 \text{ m}$ west, Andi's total displacement is zero. This total displacement can be represented as the zero vector ( $\vec{0}$ ).

Equivalent Vectors (Equal Vectors)

Two or more vectors are said to be equivalent or equal if they have the same magnitude (length) and direction, even if their starting points are different.

Consider the graph below showing three equivalent vectors: $\overrightarrow{CD}$ , $\overrightarrow{EF}$ , and $\overrightarrow{KL}$ .

Equivalent Vectors

Vectors

CD

,

EF

, and

KL

have the same magnitude and direction.

The three vectors above have the same magnitude and direction, so they are equivalent. We can write this as:

\overrightarrow{CD} = \overrightarrow{EF} = \overrightarrow{KL}

A vector is said to be equivalent to another vector if it has the same magnitude and direction as the other vector.

Exercise

Consider the two vectors below:

Comparison of Vector

A

and Vector

100 \text{ m}

Comparison of Vector

A

and Vector

B

Is vector $\vec{A}$ the opposite vector of $\vec{B}$ ?

Answer:

Vector $\vec{A}$ is not the opposite vector of $\vec{B}$ . To be an opposite vector, two conditions must be met:

They must have the same magnitude: Visually, $|\vec{A}| \neq |\vec{B}|$ (their lengths appear different).
Their directions must be exactly opposite ( $180^\circ$ apart): The direction of $\vec{A}$ is not opposite to the direction of $\vec{B}$ .

Since these two conditions are not met, $\vec{A}$ is not the opposite vector of $\vec{B}$ .

How to make vectors $\vec{A}$ and $\vec{B}$ opposite?

To make vectors $\vec{A}$ and $\vec{B}$ opposite, you could define them like this:

Opposite Vectors

A

and

B

Vector

A

and

B

are opposite

Vector Subtraction

Zero Vector

\vec{A}

-\vec{A}

180^\circ