Vector Addition

Given vectors $A = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 \\ 2 \\ 7 \end{bmatrix}$, compute $A + B$.

Vector addition is a fundamental concept in linear algebra and is essential for understanding more complex vector operations and linear transformations. When adding two vectors, such as the given vectors A and B, you are essentially combining the corresponding components of each vector. The result is a new vector that represents a translation in space, defined by the summation of each component. This operation illustrates how vectors can be visually and numerically manipulated to yield new directional information, which is crucial in physics and engineering applications, such as determining net force or velocity.

Conceptually, vector addition is commutative, meaning that the order in which vectors are added does not matter, and it is associative, which allows the grouping of vectors to be changed without affecting the result. Understanding these properties is important because they underscore the flexibility and utility of vector addition in mathematical modeling and physical simulations. Furthermore, learning how to perform operations like this by hand strengthens one's ability to interpret and manage multi-dimensional data practically, which is a skillset often used in data science and machine learning contexts.