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

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra that takes two vectors and returns a scalar. It is calculated by multiplying the corresponding components of the vectors and summing the results. For vectors u = ai + bj and v = ci + dj, the dot product is given by u ⋅ v = ac + bd. Understanding this concept is crucial for solving the given problem, as it involves calculating the dot products of the vectors u and v, and u and w.

Vector Components

Vectors can be expressed in terms of their components along the coordinate axes. In this case, the vectors u, v, and w are represented in a two-dimensional Cartesian coordinate system as u = 2i - j, v = 3i + j, and w = i + 4j. Recognizing the components of each vector allows for straightforward calculations of operations like the dot product, as each component corresponds to a specific direction in the plane.

Scalar Addition

Scalar addition involves combining scalar quantities to produce a single scalar result. In the context of the problem, after calculating the dot products u ⋅ v and u ⋅ w, the next step is to add these two scalar results together. This concept is essential for arriving at the final answer, as it requires a clear understanding of how to manipulate and combine scalar values derived from vector operations.