Subtracting Two Vectors

Subtract the vector $Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix}$ from $X = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$. Compute $X - Y$.

When faced with the problem of subtracting two vectors, it's essential to understand the basic operations involving vectors which are foundational in linear algebra. Vectors are mathematical objects characterized by both a direction and a magnitude, and they can be used to represent a variety of physical quantities, such as forces or velocities, and abstract mathematical constructs. In vector arithmetic, subtraction can be viewed graphically as the addition of the negative of a vector, essentially flipping the direction of the vector being subtracted and then performing vector addition.

The task of subtracting vector Y from vector X requires the operation to be done component-wise. This means you need to subtract the corresponding components of Y from those of X. This problem helps in strengthening your understanding of elementary vector operations, which you would apply in more complex scenarios such as in physics for finding resultant forces or in computer graphics for transformations. The principles you practice here form the bedrock upon which more complex topics like linear combinations and vector spaces are built.

In practice, being adept at these operations contributes to your problem-solving skills across several domains, including engineering disciplines, computational fields, and applied sciences. The operation of subtraction and addition of vectors can be expanded into higher dimensions where more components are involved, thus broadening the applicability of these fundamental concepts.