Zero Vector (Null Vector)

A zero vector or null vector is a vector that has a zero magnitude and no direction. Before getting into the detail about the zero vector, let us first recall the meaning of vectors. A vector is an object or a geometric entity that has a magnitude (length) and direction. Now, a vector that has zero length and an undefined direction is called a zero vector or null vector. It has all components equal to 0.

In this article, we will study the concept of zero vector, its definition, and symbol and solve some examples using zero vector (null vector) for a better understanding of the concept.

A zero vector is a vector entity in an n-dimensional space that has a magnitude equal to 0 and points in no direction. Let us consider a real-life example to understand this. Suppose two people are pulling a rope from its two ends with equal force but in opposite directions. So, the net force applied to the rope will be a zero vector (null vector) as the two equal forces balance each other out because they are in opposite directions. Let us see the definition and how to represent a null vector in the following sections.

A zero vector or a null vector is defined as a vector in space that has a magnitude equal to 0 and an undefined direction. Zero vector symbol is given by \(\overrightarrow{0} = (0, 0 , 0)\) in three dimensional space and in a two-dimensional space, it written as \(\overrightarrow{0} = (0, 0)\). The components of a null vector are all equal to 0 as it has zero length and it does not point in any direction. It is also called the additive identity of the set of vectors as whenever we add a zero vector to any other non-zero vector, the resultant is equal to the given non-zero vector.

Now that we have understood the meaning of the null vector, let us go through some of its properties to understand more about it. Given below are some of the important properties of a zero vector (null vector):

• When a zero vector is added to a non-zero vector, the resultant vector is equal to the given non-zero vector, i.e., \(\overrightarrow{0} + \overrightarrow{a} = \overrightarrow{a}=\overrightarrow{a} + \overrightarrow{0}\).
• When a zero vector is subtracted from a non-zero vector, the resultant vector is equal to the given non-zero vector, i.e., \(\overrightarrow{a} - \overrightarrow{0} = \overrightarrow{a}\).
• When a non-zero vector is subtracted from a null vector, the resultant vector is equal to the negative of the given non-zero vector, i.e., \(\overrightarrow{0} - \overrightarrow{a} = \overrightarrow{-a}\).
• The dot product of a zero vector with any vector is 0. i.e., \(\overrightarrow{a}.\overrightarrow{0} = 0 =\overrightarrow{0}.\overrightarrow{a}\)
• The cross product of a null vector with any vector is always equal to the null vector. i.e., \(\overrightarrow{0} \times \overrightarrow{a} = \overrightarrow{0}=\overrightarrow{a} \times \overrightarrow{0}\)

We have understood the concept of a zero vector. Let us now see its significance in the field of vector algebra and real-life through an example. Consider a man running in the east direction for 5 km. After running for 5 km in the east direction, he runs back towards the west direction for another 5 km. East and west are in opposite directions, so after running towards the west for 5 km, the man is at the point where he started. So, technically he has covered 10 km of distance by running but in vector algebra, his displacement is considered to be 0 km as his final and initial points are the same.

Displacement is a vector quantity whereas distance is a scalar quantity. So, the significance of the zero vector is that even though the displacement is a null vector but it tells the arbitrary direction (west) in which the man is facing.

Important Notes on Zero Vector:

• A zero vector has no length and does not point in any specific direction.
• A null vector is an additive identity in vector algebra.
• The resultant of the product of zero vector with any other vector is always zero.

☛ Related Topics:

• Adding Vectors Calculator
• Resultant Vector Calculator
• Angle Between Two Vectors Calculator

What is Zero Vector in Vector Algebra?

A zero vector or a null vector is a geometrical entity in an n-dimensional space that has a magnitude equal to 0 and points in no direction. It has all components equal to 0. It is one of the types of vectors.

What is a Non-Zero Vector?

A non-zero vector is a vector with a non-zero magnitude. A non-zero vector can have a component equal to zero but all its components cannot be zero at the same time.

Why is a Set of Vectors with Zero Vector Linearly Dependent?

By definition, any set of vectors that contain a zero vector is linearly dependent. This is because all scalars of the linear combination of these vectors will not necessarily be zero which will make it a non-trivial linear combination of vectors. Hence the set of vectors with zero vector becomes linearly dependent.

How do you Identify a Zero Vector?

If all the components of a vector are equal to 0, then the vector is a zero vector. Also, the null vector has a magnitude equal to 0 and points in no direction.

Is Null Vector Unique?

Yes, a null vector is unique as no other vector can have all components equal to zero other than the null vector.

How to Write Null Vector?

Zero vector or null vector symbol is given by \(\overrightarrow{0} = (0, 0 , 0)\) in three dimensional space and in a two-dimensional space, it written as \(\overrightarrow{0} = (0, 0)\).