Calculate Dot Product of Scaled Vector

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

The dot product is an important operation in vector algebra, providing insights into the relationship between two vectors in terms of their magnitude and direction. In this problem, you are not only asked to compute the dot product of two vectors but also to consider the effect of scaling one of these vectors by a scalar. Scaling a vector by a scalar changes its magnitude but not its direction, magnifying or shrinking it by the factor of the scalar in each component. This transformation must be considered when calculating the dot product, which is a sum involving products of the corresponding components of the two vectors in question.

To solve this problem, you need to first scale vector 'a' by multiplying each of its components by 3, resulting in a new vector. The next step is to apply the dot product formula: multiply the corresponding components of vectors 'b' and the scaled 'a', and then sum these products. The resulting scalar value gives a measure of how much one vector extends in the direction of the other. Understanding dot product is a crucial foundation for exploring more complex vector operations like projection and cross product, and has applications in physics, engineering, and computer graphics.

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