Operations on vectors (addition, scalar multiplication)

Vectors can undergo two primary operations: addition and scalar multiplication. These operations allow for the manipulation and combination of vectors. Here's a brief explanation of each operation:

1. Addition of Vectors:

   Vector addition is performed by adding corresponding components of two or more vectors. For example, consider two vectors, u = [u₁, u₂, u₃] and v = [v₁, v₂, v₃]. The sum of these vectors, denoted as u + v, is calculated by adding their corresponding components:

   u + v = [u₁ + v₁, u₂ + v₂, u₃ + v₃].

   The result is a new vector with components obtained by adding the corresponding components of the original vectors.

2. Scalar Multiplication of Vectors:

   Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation scales the vector by the given scalar. To perform scalar multiplication, each component of the vector is multiplied by the scalar. For instance, consider a vector v = [v₁, v₂, v₃] and a scalar c. The scalar multiplication of c * v is calculated as:

   c * v = [c * v₁, c * v₂, c * v₃].

   Each component of the vector is multiplied by the scalar, resulting in a new vector that is scaled by the scalar value.

These operations on vectors follow certain properties:

1. Commutative Property of Addition:

   u + v = v + u

   The order of addition does not affect the result.

2. Associative Property of Addition:

   (u + v) + w = u + (v + w)

   The grouping of vectors in addition does not affect the result.

3. Distributive Properties:

   a * (u + v) = a * u + a * v

   Scalar multiplication distributes over vector addition.

   (a + b) * u = a * u + b * u

   Addition of scalars distributes over scalar multiplication of vectors.

These properties make vector addition and scalar multiplication versatile operations for manipulating and combining vectors in various mathematical applications, such as linear algebra, physics, and computer graphics.