Question

What happens when you add a vector to a zero vector?

Answer 1

Hint: We define a vector as an object with a length and a direction. However, there is one exception to vectors having a direction: the zero vector, i.e., the unique vector having zero length. With no length, the zero vector is not pointing in any particular direction, so it has an undefined direction. So we need to carry out the addition of a null vector to another vector with a certain magnitude and direction.

Complete step by step answer:
A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors could be added together to define the result or resultant.
$Vector = Magnitude \times Direction$
For Zero vector $Magnitude = 0$but it has direction
\[\left| {\overrightarrow B } \right| = 0\](Zero or Null vector)
\[\overrightarrow R \,\, = \,\,\overrightarrow A + \left( {\overrightarrow B } \right)\]
\[\overrightarrow R \,\, = \,\,\overrightarrow A + \left( {\overrightarrow 0 } \right)\] From the addition of vectors.
Therefore $\overrightarrow R = \,\overrightarrow A $
A zero vector which is also known as a null vector is a vector that has an arbitrary direction and has zero magnitudes. If a vector is added to a zero vector, the resulting vector is the same as the vector that was added to the zero vector.
 For example, if vector A is added to a zero vector, the resulting vector would be the same as vector A. Likewise, when a zero vector is subtracted from a vector, the resulting vector is the same as the one from which the zero vector was subtracted. Also, it can be noted that any non-zero scalar, when multiplied with a zero vector, also yields a zero vector. Lastly, it can also be distinguished that any vector 3multiplied by zero yields a zero vector.

Note:
There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods are a) Pythagorean Theorem and trigonometric methods b) the head-to-tail method using a scaled vector diagram. If we have an arbitrary number of dimensions, the zero vector is the vector where each component is zero.