Dot Product of Vectors

Calculate the dot product of vectors $\mathbf{a} = (3, -4, 7)$ and $\mathbf{b} = (5, 2, -3)$.

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the context of vectors in three-dimensional space, the dot product of two vectors can be computed using the sum of the products of their corresponding components. In more straightforward terms, if you have two vectors a and b, represented as (a1, a2, a3) and (b1, b2, b3) respectively, their dot product is calculated as $a_1*b_1 + a_2*b_2 + a_3*b_3$. This operation not only gives a measure of parallelism between two vectors but also has a geometric significance. Specifically, the dot product relates to the cosine of the angle between the two vectors. If the dot product is zero, it implies that the vectors are orthogonal, meaning they are at right angles to one another. When attempting to determine the dot product, understanding this geometric relationship can provide deeper insight into the alignment and orientation of vectors in space. Moreover, the dot product is a foundational concept in various mathematical and physical applications, including projections, work calculations in physics, and determinations of vector orthogonality.

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