Question

State following as true or false:
Zero vector is unique.

Answer 1

Hint: Prove it by contradiction.
Assume that there are two unique vectors. Then arrive at a contradiction.

Any quantity which has both magnitude and direction is called a vector quantity.

Complete step-by-step answer:

Zero vector or null vector is a vector which has zero magnitude and an arbitrary direction.
It is represented by $\overrightarrow{0}$.
The main two properties of zero vector are, if we multiply the zero vector by any other non zero vector we will get zero vector.
If you add zero vector with any other vector the other vector remains the same.
That means, let $\overrightarrow{a}$ be a non zero vector. If we add the zero vector with this we will get $\overrightarrow{a}$.
$\overrightarrow{a}+\overrightarrow{0}=\overrightarrow{0}+\overrightarrow{a}=\overrightarrow{a}$.
Similarly if we subtract a zero vector from a non zero vector the non vector remains the same.
It is important to note that we cannot take the above result to be a number, the result has to be a vector.
For example we can say, the position vector of the origin of the coordinate axes is a zero vector.
Now in this question we have to check if the zero vector is unique or not.
We will prove that by contraction.
Let us assume that there are two zero vectors, say, $\overrightarrow{0}$ and $\overrightarrow{{{0}_{1}}}$.
Let us take a non zero vector say, $\overrightarrow{a}$.
Now according to the property of the zero vector we can say that:
$\begin{align}
  & \overrightarrow{a}+\overrightarrow{0}=\overrightarrow{a}......(1) \\
 & \overrightarrow{a}+\overrightarrow{{{0}_{1}}}=\overrightarrow{a}.....(2) \\
\end{align}$
Now, we can equate (1) and (2)
$\overrightarrow{a}+\overrightarrow{0}=\overrightarrow{a}+\overrightarrow{{{0}_{1}}}$
We can cancel out the $\overrightarrow{a}$ from both the sides.
$\overrightarrow{0}=\overrightarrow{{{0}_{1}}}$
The two zero vectors are equal, which is a contraction to our assumption.
Hence, the zero vector is unique.
Therefore the given statement is true.

Note: In vector space zero vector is always defined with respect to addition not with respect to multiplication. We generally make the mistake here, in a vector space the identity element with respect to addition is known as the zero vector and the identity element with respect to multiplication is known as unity.