Given vectors \(x, y, z \in \mathbb{R}^{n}\) and the scalar \(a \in \mathbb{R}\), prove the following identities: 1\. \(x+y=y+x\). Is vector addition commutative?

The commutative property is a fundamental principle in mathematics. It simply means that changing the order of the operands does not change the result. This applies to basic operations like addition and multiplication.
For example, with real numbers: \(a + b = b + a\) and \(a \times b = b \times a\).
In the context of this exercise, we're focusing on the addition aspect of the commutative property.
When we talk about vector addition, ensuring that the operation is commutative allows us to switch the order of the vectors and still get the same result.
This property is vital in various fields, including physics and engineering, where vectors play a crucial role.
In our exercise, showing that \(x + y = y + x\) for any vectors \(x\) and \(y\) demonstrates this simplicity and universality in vector operations.

Vectors are entities that have both magnitude and direction, and they are used in many branches of science and engineering to represent anything from force to velocity.
Basic operations with vectors include addition, subtraction, and scalar multiplication.
Vector addition is done component-wise, meaning each corresponding pair of components from the vectors is added together.
If you have two vectors \(x = (x_1, x_2, ..., x_n)\) and \(y = (y_1, y_2, ..., y_n)\), the result of their addition is another vector \(x + y = (x_1 + y_1, x_2 + y_2, ..., x_n + y_n)\).
In simpler terms, add the first components of both vectors, then the second components, and so on, up to the \(n\)-th components. This operation is fundamental in many practical applications, like adding forces in physics.
Performing these vector operations accurately is crucial for solving numerous real-world problems efficiently.

To prove that vector addition is commutative, let's follow some steps.
Let's start with the definition of vector addition. Vectors \(x\) and \(y\) have components \(x = (x_1, x_2, ..., x_n)\) and \(y = (y_1, y_2, ..., y_n)\), respectively.
Using the definition of vector addition: \(x + y = (x_1 + y_1, x_2 + y_2, ..., x_n + y_n)\).

Now, consider reversing the order of addition. This means computing \(y + x\): \(y + x = (y_1 + x_1, y_2 + x_2, ..., y_n + x_n)\).
Here, we see that the components are the same as those in \(x + y\) but written in reverse order.
Next, we use the commutative property of real numbers, where \(a + b = b + a\). For each component, \(x_i + y_i = y_i + x_i\).
Applying this to all components, we easily see that \( (x_1 + y_1) = (y_1 + x_1) \), \( (x_2 + y_2) = (y_2 + x_2) \), and so forth, holds true for all \(n\) components.
Therefore, \( x + y = y + x \), proving that vector addition is indeed commutative in nature.