Negative Vector or Negative of a Vector is vector with same magnitude but in opposite direction. The negative of a vector is obtained by multiplying the vector by -1. This operation changes the direction of the vector while keeping its magnitude the same. Essentially, if you have a vector [Tex]\vec{a}[/Tex], its negative vector −[Tex]\vec{a}[/Tex] will point in the exact opposite direction but will still have the same length as [Tex]\vec{a}[/Tex].

In this article, we will discuss Negative of a Vector in detail including how to find negative vector.

Definition of a Vector

A vector is a mathematical entity that has both magnitude (length) and direction.

Vectors are often represented as arrows in a coordinate system, where the length of the arrow indicates the vector’s magnitude and the direction of the arrow shows the direction in which the vector points.

Types of Vectors

There are many types of vector based on different properties and relations:

• Zero Vector
• Unit Vector
• Position Vector
• Equal Vector
• Negative Vector
• Co-Initial Vectors
• Collinear Vectors
• Parallel Vectors
• Orthogonal Vectors

Negative Vector

A negative vector is a vector that has the same magnitude as a given vector but points in the opposite direction. If the original vector is [Tex]\vec{v}[/Tex], its negative is denoted by [Tex]-\vec{v}[/Tex].

• Magnitude: The magnitude of [Tex]-\vec{v}[/Tex] is equal to the magnitude of [Tex]\vec{v}[/Tex].
• Direction: The direction of −[Tex]\vec{v}[/Tex] is opposite to the direction of [Tex]\vec{v}[/Tex].

Representation of Negative of a Vector

If a vector [Tex]\vec{v}[/Tex] is represented in component form as: [Tex]\vec{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}[/Tex]

Then, the negative vector [Tex]-\vec{v}[/Tex] is:

[Tex] -\vec{v} = \begin{pmatrix} -v_x \\ -v_y \end{pmatrix}[/Tex]

Example of Negative Vector

Some of the examples of negative vector and their original vectors are:

• Original Vector: [Tex] \vec{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}[/Tex]
• Negative Vector: [Tex]-\vec{a} = \begin{pmatrix} -2 \\ -5 \end{pmatrix}[/Tex]

• Original Vector: [Tex]\vec{b} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}[/Tex]
• Negative Vector: [Tex]-\vec{b} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}[/Tex]

• Original Vector: [Tex]\vec{c} = \begin{pmatrix} 0 \\ 7 \end{pmatrix}[/Tex]
• Negative Vector: [Tex]-\vec{c} = \begin{pmatrix} 0 \\ -7 \end{pmatrix}[/Tex]

• Original Vector: [Tex]\vec{d} = \begin{pmatrix} -1 \\ -1 \end{pmatrix}[/Tex]
• Negative Vector: [Tex]-\vec{d} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}[/Tex]

How to Find Negative of a Vector?

Finding the negative of a vector involves reversing the direction of the vector while maintaining its magnitude. Here’s how you can do it step by step:

Step 1: Identify the components of the original vector.

Suppose you have a vector [Tex]\vec{v}[/Tex] with components v_x, v_y, and v_z in three-dimensional space. The vector can be written as:

[Tex]\vec{v} = \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix}[/Tex]

Step 2: Negate each component:

Multiply each component of the vector by -1 to get the components of the negative vector:

[Tex]\vec{v} = \begin{pmatrix} -v_x \\ -v_y \\ -v_z \end{pmatrix}[/Tex]

Let’s consider an example for better understanding:

Example 1: Find the negative vector of [Tex]\vec{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}[/Tex].

Solution:

Step 1: Identify the components 3 and 4.

Step 2: Negate each component.

Thus, [Tex]-\vec{a} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}[/Tex]

Example 2: Find the negative vector of [Tex]\vec{b} = \begin{pmatrix} -2 \\ 5 \\ 7 \end{pmatrix}[/Tex].

Solution:

Step 1: Identify the components -2, 5, and 7.

Step 2: Negate each component:

Thus, [Tex]-\vec{b} = \begin{pmatrix} 2 \\ -5 \\ -7 \end{pmatrix}[/Tex]

Properties of Negative Vectors

Some of the common properties of negative vectors are:

• The magnitude (length) of a vector and its negative are the same i.e., [Tex]|\vec{v}| = |-\vec{v}|[/Tex].
• The direction of a negative vector is exactly opposite to that of the original vector.
• The sum of a vector and its negative is the zero vector i.e., [Tex]\vec{v} + (-\vec{v}) = \vec{0}[/Tex]
• Subtracting a vector is equivalent to adding its negative.

[Tex]\vec{u} – \vec{v} = \vec{u} + (-\vec{v})[/Tex]

• Multiplying a vector by -1 gives its negative.

[Tex]-1 \cdot \vec{v} = -\vec{v}[/Tex]

• The dot product of a vector with its negative is the negative of the dot product of the vector with itself.

[Tex]\vec{v} \cdot (-\vec{v}) = -(\vec{v} \cdot \vec{v})[/Tex]

• The cross product of a vector with its negative is the zero vector.

[Tex]\vec{v} \times (-\vec{v}) = \vec{0}[/Tex]

Applications of Negative Vectors

Some of the common applications involving negative of any vector are:

• Force and Motion: Negative vectors are used to represent forces acting in opposite directions. For example, in mechanics, the reaction force is often the negative vector of the applied force.

• Electric Fields: In electrostatics, the direction of the electric field vector points away from positive charges and toward negative charges. The field due to a negative charge is the negative of the field due to a positive charge.

• Structural Analysis: Negative vectors are used to analyze stresses and strains in materials. Forces that act in opposite directions on structural elements are represented using negative vectors.

• Object Transformations: In 3D graphics, transformations such as translation, rotation, and scaling can involve negative vectors. For example, reversing the direction of movement or rotation can be achieved using negative vectors.

• Vector Representation of Trends: In economic modeling, vectors can represent trends and directions of financial data. Negative vectors are used to indicate opposite trends or corrections.

Conclusion

In conclusion, understanding the negative of a vector is a fundamental aspect of vector mathematics. By multiplying a vector by -1, we reverse its direction while maintaining its magnitude. This simple yet powerful operation is crucial for various vector arithmetic operations, especially in fields like physics and engineering where vectors are used to represent physical quantities.

FAQs on Negative of a Vector

Define negative vector.

A negative vector is a vector that has the same magnitude as a given vector but points in the opposite direction. If the original vector is [Tex]\vec{v}[/Tex], its negative is denoted as [Tex] -\vec{v}[/Tex].

How do you find the negative of a vector?

To find the negative of a vector, simply negate each of its components. For a vector [Tex]\vec{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}[/Tex], the negative vector is [Tex] \vec{v} = \begin{pmatrix} -v_x \\ -v_y \end{pmatrix}[/Tex].

What is the significance of a negative vector?

The negative vector represents the same magnitude as the original vector but in the opposite direction. This is useful in various applications, such as reversing forces, directions, or movements in physics, engineering, and computer graphics.

What is the relationship between a vector and its negative?

The relationship between a vector [Tex]\vec{v}[/Tex] and its negative − [Tex]\vec{v}[/Tex] is that they have the same magnitude but point in opposite directions. Adding a vector to its negative results in the zero vector: [Tex] \vec{v} + (- \vec{v}) = \vec{0}[/Tex]

Can a negative vector be zero?

Yes, if the original vector is a zero vector (null vector), [Tex]\vec{0}[/Tex], then its negative is also a zero vector, [Tex]-\vec{0}=\vec{0}[/Tex].

How are negative vectors used in physics?

In physics, negative vectors are used to represent forces, velocities, or accelerations that act in opposite directions. For example, if a force [Tex]\vec{F}[/Tex] acts to the right, the reaction force [Tex]-\vec{F}[/Tex] acts to the left.

How does negating a vector affect its magnitude?

Negating a vector does not change its magnitude.