Problem 16 - Algebra and Trigonometry

Question

15-18 Find the indicated quantity, assuming \(\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}-3 \mathbf{j},\) and \(\mathbf{w}=3 \mathbf{i}+4 \mathbf{j}\). $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})$$

Step-by-Step Solution

Answer

The result of \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})\) is 9.

Step 1: Find the Vector Sum \(\mathbf{v} + \mathbf{w}\)

To find \(\mathbf{v} + \mathbf{w}\), we need to add the corresponding components of \(\mathbf{v}\) and \(\mathbf{w}\). \(\mathbf{v} = \mathbf{i} - 3\mathbf{j}\) and \(\mathbf{w} = 3\mathbf{i} + 4\mathbf{j}\). Adding these gives: \(\mathbf{v} + \mathbf{w} = (1 + 3)\mathbf{i} + (-3 + 4)\mathbf{j}\). Simplifying this, we get \(\mathbf{v} + \mathbf{w} = 4\mathbf{i} + \mathbf{j}\).

Step 2: Calculate the Dot Product \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})\)

Now that we have \(\mathbf{v} + \mathbf{w} = 4\mathbf{i} + \mathbf{j}\), we can calculate the dot product with \(\mathbf{u}\). The vector \(\mathbf{u}\) is given by \(2\mathbf{i} + \mathbf{j}\). The dot product is calculated as \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = (2\mathbf{i} + \mathbf{j}) \cdot (4\mathbf{i} + \mathbf{j}) = 2 \times 4 + 1 \times 1\). This simplifies to \(8 + 1 = 9\).

Key Concepts

Dot Product

The dot product is a fundamental concept in vector algebra that helps us measure the degree to which two vectors point in the same direction. It's a scalar product, meaning it results in a single number rather than another vector. Calculating the dot product is straightforward: you just multiply the corresponding components of two vectors and then sum them up.

In symbols: given two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as:

• First, multiply the \(\mathbf{i}\) components: \( a_1 \times b_1 \)
• Next, multiply the \(\mathbf{j}\) components: \( a_2 \times b_2 \)
• Add these together: \( a_1b_1 + a_2b_2 \)

For example, using vectors from our original exercise, with \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \) and \( \mathbf{v} + \mathbf{w} = 4\mathbf{i} + \mathbf{j} \), we have:

• First pair: \( 2 \times 4 = 8 \)
• Second pair: \( 1 \times 1 = 1 \)
• Sum them: \( 8 + 1 = 9 \)

The result, 9, represents the dot product of \( \mathbf{u} \text{ and } (\mathbf{v} + \mathbf{w}) \). The calculation is simple yet powerful, offering insight into the relationship between the directional tendencies of vectors.

Vector Addition

Vector addition is a crucial vector operation, allowing us to combine two or more vectors into a single resultant vector. This is performed by simply adding the corresponding components of each vector.

To illustrate, let's consider two vectors, \( \mathbf{v} = \mathbf{i} - 3\mathbf{j} \) and \( \mathbf{w} = 3\mathbf{i} + 4\mathbf{j} \):

• For the \(\mathbf{i}\) components: add them together \( 1 + 3 = 4 \)
• For the \(\mathbf{j}\) components: add them together \( -3 + 4 = 1 \)

This results in the vector \( \mathbf{v} + \mathbf{w} = 4\mathbf{i} + \mathbf{j} \). This operation is foundational because it allows us to see how different forces, directions, or motions can combine to create a new, sum vector that incorporates all inputs. Vector addition is not only basic but also essential in fields like physics, computer graphics, and engineering where combining directions and magnitudes is part of solving larger problems.

Algebraic Vectors

Algebraic vectors are a symbolic representation of vectors using letters and numbers to denote direction and magnitude. This type of expression is incredibly useful for calculations involving vector operations.

In our study of algebraic vectors:

• Each vector has components represented typically as symbols \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), which correspond to the unit vectors in 3D space, or just \( \mathbf{i} \) and \( \mathbf{j} \) in 2D space.
• The components are numerical coefficients that define the vector's magnitude in both the horizontal and vertical directions.

For example, \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \) specifies a vector that moves with a magnitude of 2 in the \(\mathbf{i}\) direction and 1 in the \(\mathbf{j}\) direction. Manipulating algebraic vectors involves straightforward mathematical operations like addition and dot products, making them a preferred method when analyzing and computing vector relationships.

Understanding algebraic vectors is crucial for anyone dealing with vector-related problems because they provide a clear and concise way to interpret movement and force directions. As you get comfortable with these notations, solving complex real-world problems becomes markedly simpler.