Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the dot product of u and v. $$\begin{aligned} &\mathbf{u}=\langle-4,1\rangle\\\ &\mathbf{v}=\langle 5,-4\rangle \end{aligned}$$

Step-by-Step Solution

Answer

-24

Step 1: Identifying the vectors

Given vectors are \(\mathbf{u}=\langle-4,1\rangle\) and \(\mathbf{v}=\langle 5,-4\rangle\). So, \(u_1 = -4, u_2 = 1\) and \(v_1 = 5, v_2 = -4\).

Step 2: Calculate the dot product

Use the formula for dot product: \(u \cdot v = u_1v_1 + u_2v_2 = (-4)*(5) + 1*(-4) = -20 - 4 = -24.\)

Key Concepts

Vector Operations

When performing vector operations, it's essential to understand the fundamental properties and types of vectors. In this context, the dot product is a crucial operation often used in physics and engineering. The dot product is a way to merge two vectors to produce a scalar. Here are key points about this operation:

• Definition: The dot product (also called the scalar product) of two vectors results in a scalar quantity, not a vector.
• Components: Each vector is defined with specific components, denoted as \( \mathbf{u} = \langle u_1, u_2, \ldots, u_n \rangle \) and \( \mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle \).
• Formula: The dot product is calculated by multiplying the corresponding components of the vectors and adding the results: \( \mathbf{u} \cdot \mathbf{v} = u_1\cdot v_1 + u_2\cdot v_2 + \ldots + u_n\cdot v_n \).
• Geometric Interpretation: The dot product represents the product of the magnitudes of the two vectors and the cosine of the angle between them.

Understanding these basics aids in calculations and application across various fields, ensuring effective operations.

Algebraic Expressions

Algebraic expressions are vital in describing mathematical situations symbolically. They enable us to solve for unknowns and perform tasks like calculating the dot product. Here's how algebraic expressions are utilized in this scenario:

• Representation: Vectors are expressed in angled brackets containing their components, a standard algebraic form: \( \mathbf{u} = \langle u_1, u_2 \rangle, \mathbf{v} = \langle v_1, v_2 \rangle \).
• Simplification: Algebraic expressions allow for simplifying components through basic arithmetic operations, such as multiplication and addition: \( u_1v_1 + u_2v_2 \).
• Substitution: Substituting the known values of each vector component is a critical step, turning an abstract expression into a concrete numerical result.

Understanding how to manipulate these algebraic expressions gives students the tools to handle a broad array of mathematical problems beyond just dot products.

Mathematical Calculations

Performing mathematical calculations accurately is crucial for finding the correct dot product. In the exercise solution, we see a practical example of such calculations applied to vectors. Here's an insight:

• Basic Operations: The exercise involves multiplication and addition, the core arithmetic operations fundamental in calculations.
• Execution: Each step in the calculation should be carefully carried out to avoid errors:\((-4) \times (5) = -20\) followed by:\(1 \times (-4) = -4\).
• Summation: Adding the results of these multiplications gives the final dot product: \( -20 + (-4) = -24\).
• Check Your Work: Always review each step to ensure accuracy; this prevents small mistakes from leading to larger errors in the final outcome.

Being methodical and precise with mathematical calculations ensures dependable results in any mathematical task.