Calculate the Dot Product Using Scalar Multiplication

Calculate the dot product of $\mathbf{u}$ and $3\mathbf{v}$ by using a shortcut method for scalar multiplication.

The dot product, also known as the scalar product, is a central topic in vector calculus, offering a way to multiply two vectors that results in a scalar. The dot product provides significant insights, particularly in relation to projections and angles between vectors. One core concept is that the dot product between two vectors can reveal if they are perpendicular. When the dot product is zero, the two vectors are orthogonal, meaning they are at right angles to each other. This property becomes particularly valuable in various applications including physics, computer graphics, and optimization problems.

When discussing scalar multiplication, it is crucial to understand how scaling a vector impacts its magnitude while preserving its direction. In the context of vector operations, scalar multiplication is often combined with the dot product to simplify calculations. By scaling a vector before computing the dot product, one can leverage the distributive properties of the dot product to shortcut the calculation process, ultimately making the computation more efficient. This method is essential in dealing with larger vectors common in dimensions beyond two or three, where calculating individual components becomes cumbersome.

In this particular problem, the challenge involves using a shortcut for scalar multiplication to calculate the dot product of vector u and three times vector v. Understanding how scalar multiplication interacts with the distributive property of dot products helps streamline this computation. It emphasizes the importance of mastering shortcuts as these significantly enhance mathematical literacy, particularly in problem-solving settings involving higher dimensional spaces where computational efficiency is paramount.

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