In Exercises 9–16, letu = 2i - j, v = 3i + j, and w = i + 4j.Find each specified scalar.(4u) ⋅ v

Understanding vector operations is crucial in this problem. Vectors can be added, subtracted, and multiplied by scalars. In this case, the vector u is being multiplied by the scalar 4, which scales the vector's magnitude while maintaining its direction. This operation is foundational for further calculations involving dot products.

The dot product is a key operation in vector algebra that combines two vectors to produce a scalar. It is calculated by multiplying corresponding components of the vectors and summing the results. The dot product provides insights into the angle between vectors and is essential for determining orthogonality and projection.

Vectors are often expressed in component form, which involves breaking them down into their respective i (horizontal) and j (vertical) components. For example, the vector u = 2i - j has components 2 and -1. Understanding this representation is vital for performing operations like the dot product, as it allows for straightforward calculations using the individual components.