Vector Subtraction

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

Vector subtraction is an essential operation within vector algebra and forms the basis for understanding vector spaces in higher dimensions. The operation of subtracting two vectors, as presented in this problem, involves an element-wise subtraction of the vector components. Specifically, for vectors of the form A = [a1, a2, a3] and B = [b1, b2, b3], the result of B - A is given by [b1 - a1, b2 - a2, b3 - a3]. Understanding this operation is crucial for applications that involve translations in geometry and physics or solving linear algebra problems that involve vector shifts or displacements.

In a broader context, vector subtraction is closely related to the concept of vector addition and the commutative property of vectors. These operations are foundational in forming linear combinations, which are integral in solving systems of equations and understanding the structure of vector spaces. Mastering these basic vector operations allows one to explore more complex topics, such as vector projections, orthogonality, and vector transformations. This problem serves as a stepping stone to more sophisticated concepts in linear algebra and applications in various scientific domains.