Can the sum of two equal vectors be equal to either vector

When we talk about vectors in physics or mathematics, it's essential to understand their properties, especially when it comes to addition. Let's dive into your question about whether the sum of two equal vectors can be equal to either of the original vectors.

Understanding Vector Addition

Vectors are quantities that have both magnitude and direction. When we add two vectors, we essentially combine their magnitudes and directions. The sum of two vectors is found using the triangle or parallelogram method of vector addition.

Equal Vectors Defined

Two vectors are considered equal if they have the same magnitude and direction. For example, if we have vector **A** with a magnitude of 5 units pointing east, and vector **B** also has a magnitude of 5 units pointing east, then **A** and **B** are equal vectors.

Adding Equal Vectors

Now, let’s consider what happens when we add two equal vectors. If we take vector **A** and add it to itself (i.e., **A + A**), we can analyze the result:

• The magnitude of the sum will be the sum of the magnitudes: 5 units + 5 units = 10 units.
• The direction will remain the same as that of the original vectors, which is east.

Thus, the resulting vector, which we can call **C**, has a magnitude of 10 units pointing east. This means that **C** is not equal to either **A** or **B**, since both of those have a magnitude of only 5 units.

Visualizing the Concept

To visualize this, imagine two arrows of equal length pointing in the same direction. When you place them head-to-tail, the resulting arrow (the sum) will be longer than either of the original arrows. This is a fundamental property of vector addition.

Conclusion on the Equality of Vectors

In summary, the sum of two equal vectors cannot be equal to either of the original vectors. Instead, the result will always be a new vector with a magnitude that is double that of the individual vectors, maintaining the same direction. This principle is crucial in various applications, from physics to engineering, where understanding vector behavior is essential.