Question

Suppose that \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) have nonzero magnitudes. Is it possible for \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) to be zero?

Short answer

Yes, if the vectors have equal magnitudes and opposite directions.

Step-by-step solution

Understanding Vector Addition

When we add two vectors, \( \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} \), the resulting vector is a combination of the two original vectors considering both magnitude and direction. For the sum of two vectors to be zero, the two vectors must cancel each other out completely.

Condition for Zero Vector

For the vector sum \( \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} \) to be zero, the vectors must have the same magnitude but opposite directions. This means that \( \overrightarrow{\mathbf{A}} = - \overrightarrow{\mathbf{B}} \) where their directions are exact opposites and their magnitudes are equal.

Applying the Condition

Verify if \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \) can satisfy the condition \( \overrightarrow{\mathbf{A}} = - \overrightarrow{\mathbf{B}} \). Since both vectors have non-zero magnitudes, this condition can be met if they point in exactly opposite directions.

Conclusion

It is possible for the sum of two non-zero vectors \( \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} \) to be zero if \( \overrightarrow{\mathbf{A}} \) is the negative of \( \overrightarrow{\mathbf{B}} \). This is achieved when the vectors have equal magnitudes and point in opposite directions.

Key concepts

Zero Vector

A zero vector, often denoted as \( \overrightarrow{0} \), is a unique vector that results when the sum of two vectors cancels out their effects entirely. This means when two vectors are added together, and the outcome is a zero vector, their combined influence in terms of magnitude and direction results in no movement or change.
To understand this in the context of vector addition, consider adding vectors \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \). For their sum \( \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} \) to equal the zero vector, they must be of equal magnitude but in opposite directions.
In essence, the zero vector acts like the number zero in normal arithmetic – it represents a neutral element in vector addition.

Opposite Direction

When we say two vectors are in opposite directions, we mean that if one vector points in a positive direction along a given axis, the other points in the negative direction along the same axis. For example, if \( \overrightarrow{\mathbf{A}} \) points east, \( \overrightarrow{\mathbf{B}} \) would need to point west to be in the opposite direction.
Vectors that are in opposite directions counteract each other's effects when added together. This can be visualized as pushing something forward with one hand and pulling it back with the other hand simultaneously with the same force. If the magnitudes are equal, these forces (vectors) will neutralize, leading us to their vector sum being zero.
This characteristic is crucial for determining if two vectors can result in a zero vector when added.

Magnitude Equality

Magnitude, in the context of vectors, refers to the length or size of the vector. For two vectors to have magnitude equality, they must have the exact same magnitude regardless of their direction.
In mathematical terms, if vector \( \overrightarrow{\mathbf{A}} \) has a magnitude \( |\overrightarrow{\mathbf{A}}| \) and vector \( \overrightarrow{\mathbf{B}} \) has a magnitude \( |\overrightarrow{\mathbf{B}}| \), magnitude equality means \( |\overrightarrow{\mathbf{A}}| = |\overrightarrow{\mathbf{B}}| \). This implies that if these two vectors have opposite directions, based on equal magnitudes, their resulting vector sum can be zero.
Magnitude equality is a fundamental requirement for achieving a zero vector in addition and plays a pivotal role in balancing the influence of two opposing vectors.