Dot Product of a Vector Sum

Calculate the dot product between vector $\mathbf{a} = (4, 5)$ and the sum of vectors $\mathbf{b} = (3, -6)$ and $\mathbf{c} = (-8, 2)$.

In this problem, you are asked to calculate the dot product of a vector with the sum of two other vectors. This involves combining concepts from vector addition and the dot product operation. The dot product, also known as the scalar product, is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is used frequently in physics and engineering to find angles between vectors or to project one vector onto another. Understanding the dot product is crucial for interpreting geometrical properties like orthogonality and projections.

The process of solving this problem firstly requires the sum of vectors which is straightforward, involving component-wise addition of the vectors involved. Once the sum is identified, the dot product is calculated between this resultant vector and the initial vector provided. The dot product is calculated by multiplying corresponding components of the two vectors and then summing those products.

This type of problem reinforces the importance of vector operations, showing how different operations can be combined to solve complex geometric and physics-based problems. Moreover, understanding how to manipulate vectors and compute related quantities extends to many practical real-world applications, from navigation systems to computer graphics, so mastery of these concepts forms a fundamental part of studying mathematics and physics.

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