A zero vector, also known as a null vector, has a magnitude of 0 and no direction. To understand the zero vector in further depth, let us first review what a vector is and what it means to be zero. Generally speaking, a vector is a geometric item or object with both a magnitude (length) and a direction. A vector with 0 lengths and an indeterminate direction is now referred to as a zero vector or a null vector, respectively. This formula has all of its components equal to 0.

What is a Zero Vector?

A zero vector is a vector item in n-dimensional space with a magnitude equal to zero and points in no direction. Consider the following real-life scenario to comprehend better what I’m talking about. Suppose two persons are tugging on the same end of a rope with equal power but in different directions from its two ends. In this case, the net force exerted on the rope will be a zero vector (null vector) since the two equal forces cancel each other out because they act in opposite directions.

Importance of the Null Vector

We have grasped the idea of a vector with zero vector. Let us now consider the relevance of this concept in the field of vector algebra and real-life by way of an example. Consider the case of a man who runs 5 kilometres in the east direction. After running for 5 kilometres in the east direction, he turns around and runs for another 5 kilometres in the west direction. Because east and west are in opposite directions, the guy arrives at his starting position after sprinting for 5 kilometres in the west direction. As a result, he has theoretically travelled 10 kilometres via running. Still, in vector algebra, his displacement is deemed to be zero kilometres because his end and beginning positions are the same.

• Displacement is measured as a vector quantity, whereas distance is calculated as a scalar measurement. In other words, even if the displacement vector is zero, the zero vector indicates that the individual is facing in an arbitrary direction (west), which is of relevance.
• In the actual world, the number zero exists. Zero, on the other hand, means nothing. A vector with zero value and direction exists in the same way, but it does not have either direction or value.
• The starting point and ending point are not different in this situation, and they both meet at the same position logically. As a result, using null vectors in vector algebra is required.

Real-Life Applications of Zero vector 

A few real-world situations in which zero vector is employed consistently are listed below.

1. Consider this scenario: you travelled from California to Los Angeles for work and then returned to California in the same manner when you finished your job. Take a look at your route in this situation, and you will notice that your overall displacement is zero.
2. The train you’re on has begun travelling at a rapid rate towards its destination, and you’re sitting in the first row. Except for a few brief seconds, another train has already started its journey on the other side of the tracks from your location. Looking at the passengers on the other train, you see that its location appears to be constant. To put it another way, the relative velocity of the other train will be 0 when compared to your train. The location of the people on the second train provides you with a continual sense of security. The velocity of the first train will always be equal to that of the second train, in this situation, and vice versa.
3. Assume that a particle is in a stable state of equilibrium. As a result, because the particle travels through space at a constant speed, the particle’s acceleration will be zero.
4. An item is subjected to the equal force applied from two opposing sides along the horizontal axis. There is no full force delivered to the item in this situation. The item is subjected to a force whose net value is zero. Therefore, the entire applied force may be written as a zero-vector, as shown in the diagram.

Properties of Zero Vector

• A non-zero vector will be produced by subtracting the zero vector from any non-zero vector.

• We may add any non-zero vector to any zero vector, and the resulting vector will always be that same non-zero vector.

• The resulting vector will be a zero vector instead of a non-zero vector if we multiply a zero vector with any other non-zero vector in this case.

Points to remember

• Null point vectors have no length and do not point in any definite direction, whereas positive vectors do.
• Among the identities of vectors, null vectors are additive identities because adding null vectors with any other non-zero vector will result in the same non-zero vector.
• Every time you multiply a null vector by another vector, you get 0 as the consequence of that multiplication!
• When the coordinate axes are at their origin, the position vector of the source is a zero vector.
• The displacement of a stationary particle between any two different points of time is equal to zero.
• The displacement of a ball that has been thrown up and retrieved back by the thrower is a zero-vector displacement.
• The vector of the velocity of a stationary body equals zero in magnitude.
• During uniform motion, the acceleration vector of a body is equal to zero.

Summary

A zero vector or null vector is a vector in space with magnitude 0 and direction unknown. To write the zero vector sign in two dimensions, use the following formulas: A null vector has zero length and no direction. Hence its components are all 0. It is also known as the additive identity of the set of vectors because when a zero vector is added to another non-zero vector, the outcome is the same.