Problem 21 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The result is 9.

Step 1: Find the Dot Product of \(\mathbf{u}\) and \(\mathbf{v}\)

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is calculated as follows: \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \). Using \(\mathbf{u} = 2\mathbf{i} + \mathbf{j}\) and \(\mathbf{v} = \mathbf{i} - 3 \mathbf{j}\), we have \( u_1=2, u_2=1, v_1=1, v_2=-3 \). Therefore, \( \mathbf{u} \cdot \mathbf{v} = 2 \cdot 1 + 1 \cdot (-3) = 2 - 3 = -1 \).

Step 2: Find the Dot Product of \(\mathbf{u}\) and \(\mathbf{w}\)

To find \( \mathbf{u} \cdot \mathbf{w} \), use \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \) and \( \mathbf{w} = 3 \mathbf{i} + 4 \mathbf{j} \). The components are \( u_1=2, u_2=1, w_1=3, w_2=4 \). Thus, \( \mathbf{u} \cdot \mathbf{w} = 2 \cdot 3 + 1 \cdot 4 = 6 + 4 = 10 \).

Step 3: Sum the Results of Both Dot Products

Now add the two results obtained from previous steps: \( \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} = -1 + 10 = 9 \).

Key Concepts

Vectors

Vectors are an essential concept in mathematics and physics. They are objects that have both a magnitude and a direction. Unlike scalar quantities, which are described by a single number, vectors require multiple numbers to fully describe them.

Vectors are usually represented in a coordinate system as an ordered pair \[\textbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j}\]where \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors along the horizontal and vertical axes, respectively. These unit vectors have a magnitude of 1 and point in the positive direction of their respective axes.

The notation becomes crucial when dealing with operations like addition and subtraction, as well as when calculating dot products or cross products. Understanding vectors' components and how they interact fundamentally underpins vector operations.

Vector Operations

Vector operations are mathematical tools used to manipulate vectors. One of the basic operations includes the dot product. This operation is crucial as it provides a measure of how much one vector goes in the direction of another. It’s particularly used in determining angles between vectors and projections.

The dot product, sometimes called the scalar product, between two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) is calculated as:\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2\]This result is a scalar, a single number, making it different from more complex vector operations like the cross product.

Beyond the dot product, vector operations include addition, subtraction, and scaling, each with specific rules about how vector components are manipulated. Comprehending these operations is pivotal for applications across numerous science and engineering fields.

Precalculus

Precalculus serves as a bridge between algebra and calculus, examining functions, sequences, series, and the behaviors of mathematical entities like vectors. Understanding vectors and their operations is a fundamental part of this field, laying the groundwork for more advanced topics in calculus.

In precalculus, you'll often learn how vectors can be applied in real-world situations to model forces, demonstrate the change in position or velocity, and solve geometric problems. You get to explore how algebraic techniques extend to vectors, allowing for transformational operations and exploration of vector spaces. This exploration includes operations like dot products, which are further used in calculus for integrating vector fields.

This harmonious development from understanding basic operations to applying these concepts in complex analyses helps solidify students' mathematical foundation, preparing them for more advanced studies in calculus.