Dot Product of Vector with Itself

Compute the dot product of vector $\mathbf{u} = (3, 12)$ with itself.

In this problem, we explore the concept of the dot product by calculating it for a specific vector with itself. The dot product, also known as the scalar product, is fundamental in vector algebra. It represents an algebraic operation that returns a single number (a scalar) from two equal-length sequences of numbers, typically coordinate vectors. Calculating the dot product involves multiplying the corresponding components of the vectors and then summing those products.

When you compute the dot product of a vector with itself, a particularly meaningful result emerges: the sum equals the square of the vector's magnitude or length. This realization underscores the utility of the dot product in not only determining angles between vectors (when using it between two different vectors), but also in deriving the norm of a vector when performed on itself. Thus, solving this problem facilitates a deeper understanding of both magnitude and properties of vectors, which are key concepts in physics and engineering, particularly in fields such as mechanics and electromagnetism. Understanding how to manipulate and compute the dot product can also aid in grasping more complex vector operations later on, such as cross products and projections.

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