Zero Vector

Zero Vectors are the vectors with all its components zero. A zero vector or a null vector is defined as a vector in space with a magnitude equal to 0 and an undefined direction. The zero-vector symbol for three-dimensional space is given by = (0, 0, 0), and the zero-vector symbol for two-dimensional space is given by = (0, 0).

In this article, we will learn about zero vectors with zero vector definition, zero vector symbol, properties of zero vectors, and, the significance of zero vectors.

What is Zero Vector (Null Vector)?

Zero Vectors, as the name suggests, are vectors characterized by a magnitude of zero. In the realm of vector algebra, these special vectors hold distinctive properties that set them apart from non-zero vectors. Understanding their nature is fundamental for various mathematical applications.

Zero Vector Definition

A Zero Vector is a vector with all its components equal to zero. Mathematically, if V is a vector, it can be denoted as V = [0, 0,…, 0] in an n-dimensional space, where each component is zero.

Zero Vector Symbol

The zero vector is commonly represented by the symbol indicating its null magnitude. This symbolism is essential in mathematical notations and equations involving vectors.

Learn, Types of Vectors

Zero Vector Explanation

Consider a scenario where two individuals are engaged in a tug-of-war, each pulling a rope with an equal force but in opposite directions. In this situation, the forces applied by the two individuals cancel each other out due to their opposing directions.

Let, the force exerted by the first person can be represented as a vector F₁, and the force exerted by the second person as F₂. In this context, F₁ and F₂ are equal in magnitude but opposite in direction.

F_net = F₁ + F₂ = +F +(−F)

This simplifies to F_net = 0, indicating that the net force acting on the rope is a zero vector. In this case, the forces are perfectly balanced, and the rope experiences no acceleration or change in motion. The concept of a zero vector, or null vector, becomes evident as the forces in opposite directions nullify each other, resulting in a state of equilibrium.

Properties of Zero Vector (Null Vector)

Zero Vectors exhibit unique properties that distinguish them from non-zero vectors. Understanding these properties is crucial for manipulating vectors in mathematical expressions. Some key properties include:

Scalar Multiplication with Zero Vectors

When we multiply a vector by a number (let’s call it a scalar), it is called scalar multiplication. If we have a vector v = [0,0,0] and multiply it by any number, the result is still the zero vector.

Vector Addition with Zero Vectors

Adding zero vector in another vector is known as vector addition. When we add the zero vector to any other vector, the result is just the other vector. So, if we have v = [2,5,7] and add the zero vector to it, we still get 0. Zero vectors are like math’s version of a “neutral” element in addition.

Linear Independence

Zero vectors are crucial in determining linear independence or dependence of vectors. A set of vectors is linearly dependent if and only if at least one of the vectors is a scalar multiple of another, including the case where the scalar is zero.

Null Space in Linear Transformations

In the context of linear transformations and matrices, zero vectors are associated with the null space of a matrix. The null space consists of all vectors that, when transformed, result in the zero vector.

Application in Systems of Linear Equations

Zero vectors are instrumental in solving systems of linear equations. The presence of zero vectors in the solution space provides insights into the nature of solutions, such as unique solutions, no solutions, or infinite solutions.

Significance of Zero Vector (Null Vector)

Zero Vectors hold significance in various mathematical and scientific contexts. Their properties find applications in linear algebra, physics, computer science, and engineering. Understanding and leveraging these vectors enhance problem-solving capabilities and mathematical reasoning.

Role in Solving Systems of Linear Equations

A step-by-step method for solving a system of linear equations with the help of zero vectors:

Step 1: Understand the System

Begin by understanding the system of linear equations. For instance, consider a system of equation below:

• 2x + 3y – z = 5
• 4x – y + 2z = 8
• x + 2y + 3z = 10

Step 2: Formulate the Augmented Matrix

Write the system in the form of an augmented matrix:

Step 3: Perform Row Operations

Use row operations to manipulate the augmented matrix into a form where the coefficients are mostly zeros. Zero vectors will play a role here. Perform operations such as row switching, scaling, and adding rows to achieve this. The goal is to transform the matrix into reduced row-echelon form.

Step 4: Interpret the Result

The resulting matrix represents an equivalent system of equations. Interpret the matrix to deduce the solutions. If zero vectors are present, it may indicate either a unique solution, no solution, or infinitely many solutions, depending on the form of the matrix.

Here’s an example of a possible outcome:

This corresponds to the system x=2, y=1, and z=3.

Step 5: Check Solutions

Verify the solutions obtained by substituting them back into the original system of equations. This step ensures the correctness of the solution set.

This step-by-step method, involving zero vectors and matrix operations, offers a systematic approach to solving systems of linear equations. It showcases the integration of mathematical concepts, practical problem-solving, and the pivotal role of zero vectors in the process.

Read More About

• Vector Algebra
• Dot and Cross Product of Vectors

Zero Vector – Solved Examples

Example 1: Consider a ship moving eastward with a velocity vector V₁ = [10,0] and /s, and a strong westward wind exerting a force represented by the vector V₂ = [−10,0] m/s. The net velocity (V_net) of the ship can be calculated as:

Solution:

V_net = V₁ + V₂

V_net = [10,0] + [−10,0] = [0,0].

Here, the net velocity vector is a zero vector, indicating that the ship, despite the forces acting on it, remains stationary. This example illustrates the cancellation effect of opposing vectors resulting in a zero vector.

Example 2: Suppose a particle is subjected to two forces F₁ = [5,3] N and F₂ = [−5, −3] N. The resultant force (F_resultant) acting on the particle can be determined as:

Solution:

F_resultant = F₁ + F₂

F_resultant = [5, 3] + [−5, −3] = [0,0]

In this case, the net force is a zero vector, signifying that the forces are in equilibrium, and the particle remains at rest.

Zero Vector – Practice Questions

Q1. Given a vector P = [3, −4] find the vector Q such that P+Q=0

Q2. Suppose V = [−2, 7] is a vector. Determine the scalar k such that k⋅V=0

Q3. Consider two vectors A = [5, −3] and B = [−5, 3]. Verify if A−B results in a zero vector

Q4. A force F₁ = [8, −6] N is applied to an object, and an opposing force F₂​ = [−8, 6] N acts on it. Calculate the net force (F_net​) and interpret the result.

Q5. If M = [0, m] is a vector, find the value of m such that M+0 = M

Zero Vector (Null Vector) – FAQs

What is Meant by Zero Vectors?

A zero vector, denoted as 0, is a special point in space where all components are precisely zero.

What is Null Vector?

A null vector, also known as a zero vector, is a vector with all its components equal to zero.

Do Two Null Vectors have Same Direction?

Since, the direction of null vector is undefined hence the direction of two null vectors can’t be decided whether they are same or not

How Do You Find the Zero Vectors?

To find the zero vector, we multiply the vector from a vector space with scalar 0.

What is Difference between Zero Vector and Zero Scalar?

Zero Vector has magnitude equal to zero value in vector space that is it still has direction but undefined while zero scalar has magnitude zero and no direction