Dot Product Calculation and Vector Scaling

Calculate the dot product of $a$ and $b$ times vector $a$, where $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (5, -4)$.

In this problem, you are tasked with calculating the dot product of two vectors, a fundamental operation in vector algebra. The dot product, also known as the scalar product, is a way of multiplying two vectors to produce a scalar. This operation is widely used in physics and engineering, as it can, for example, indicate how much of one vector goes in the direction of another. In simple terms, it reflects the degree to which two vectors are parallel.

Once the dot product is determined, you scale one of the vectors by this scalar value. This step highlights the importance of understanding vector scaling, where a vector is multiplied by a scalar to adjust its magnitude without altering its direction. This concept is crucial in applications such as physics simulations or graphical transformations where vectors need to be adjusted while maintaining their inherent direction.

Through solving this problem, you reinforce your understanding of vector operations and their real-world applications, providing a solid foundation for more complex topics in vector calculus and multidimensional analysis.

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