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

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. This operation is crucial for determining the angle between vectors and for projecting one vector onto another.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number), which scales the vector's magnitude without changing its direction. If k is a scalar and v is a vector, then k * v results in a new vector whose length is k times that of v. This concept is essential when manipulating vectors in various operations, including scaling the result of the dot product.

Vector Components

Vectors in a two-dimensional space can be expressed in terms of their components along the x-axis and y-axis. For example, a vector u = ai + bj has components a and b, where 'i' represents the unit vector in the x-direction and 'j' in the y-direction. Understanding vector components is vital for performing operations like the dot product, as it allows for the straightforward application of the formula using the individual components of the vectors.