Vector Addition and Scalar Multiplication

Given two vectors, $\mathbf{v} = 2\mathbf{i} - 5\mathbf{j}$ and $\mathbf{w} = -3\mathbf{i} + 7\mathbf{j}$, perform the following operations:

(A) $\mathbf{v} + \mathbf{w}$

(B) $\mathbf{v} - \mathbf{w}$

(C) $2\mathbf{v} + 3\mathbf{w}$

(D) $4\mathbf{v} - 5\mathbf{w}$

In this problem, we explore fundamental operations involving vectors, focusing on addition, subtraction, and scalar multiplication. Vector addition involves combining two or more vectors to get another vector. This process requires aligning the vectors head-to-tail and finding the resultant vector. Mathematically, this is done by adding the corresponding components of the vectors. For instance, when adding vectors v and w, you simply add their respective i and j components to get the resultant vector.

Subtraction of vectors is similar to addition, but instead, we subtract the corresponding components. This can be visualized as adding the negative of the vector to another vector. Scalar multiplication, on the other hand, involves multiplying each component of the vector by the given scalar. This operation results in a vector that is pointing in the same direction if the scalar is positive, or in the opposite direction if the scalar is negative or inverted. It also affects the magnitude of the vector, scaling it based on the absolute value of the scalar.

Understanding these operations is crucial as they form the basis for more advanced topics in vector mathematics and are widely applicable in fields such as physics, engineering, and computer science. These foundational skills also enable the study of various vector spaces, transformations, and can be key in solving linear equations geometrically. This knowledge helps in visualizing and manipulating vectors in two-dimensional space and sets the stage for higher-dimensional vector analysis.