Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Given \(\mathbf{u}=\langle- 2,5\rangle\) and \(\mathbf{v}=\langle 4,3\rangle,\) find each vector. Do not use a calculator. $$v-u$$

Step-by-Step Solution

Answer

\( \mathbf{v} - \mathbf{u} = \langle 6, -2 \rangle \).

Step 1: Understanding Vector Subtraction

Before subtracting, recall that subtracting two vectors involves subtracting their corresponding components. So for vectors \( \mathbf{v} = \langle 4, 3 \rangle \) and \( \mathbf{u} = \langle -2, 5 \rangle \), \( \mathbf{v} - \mathbf{u} \) means \( \langle 4, 3 \rangle - \langle -2, 5 \rangle \).

Step 2: Subtract the Horizontal Components

Subtract the horizontal component of vector \( \mathbf{u} \) from the horizontal component of vector \( \mathbf{v} \): \( 4 - (-2) = 4 + 2 = 6 \).

Step 3: Subtract the Vertical Components

Subtract the vertical component of vector \( \mathbf{u} \) from the vertical component of vector \( \mathbf{v} \): \( 3 - 5 = -2 \).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to form the resultant vector. The first component is 6 and the second component is -2, so \( \mathbf{v} - \mathbf{u} = \langle 6, -2 \rangle \).

Key Concepts

Understanding Vectors

Vectors are essential tools in mathematics and can be thought of as arrows in the space, pointing in a specific direction with a certain length. They are usually expressed as an ordered pair or triplet of numbers, which are their components. For example, the vector \( \mathbf{v} = \langle 4, 3 \rangle \) represents a 2-dimensional vector. The components 4 and 3 usually correspond to directions on the x-axis and y-axis, respectively.

• Magnitude: The length of the vector; calculated using the Pythagorean Theorem for 2D vectors: \( \sqrt{x^2 + y^2} \).
• Direction: Represents the direction in which the vector points; often described using angles or vector coordinates.
• Notation: Commonly, vectors are notated by bold letters (e.g., \( \mathbf{u} \)) or with an arrow on top (e.g., \( \vec{u} \)).

Vectors serve as a foundation in many areas, including physics, engineering, and computer graphics. They are used to represent quantities that have both magnitude and direction, like velocity or force.

Component-wise Subtraction

To find the difference between two vectors, we perform component-wise subtraction. This means that we separately subtract each component of the first vector from the corresponding component of the second vector. Consider vectors \( \mathbf{v} = \langle 4, 3 \rangle \) and \( \mathbf{u} = \langle -2, 5 \rangle \). Component-wise subtraction follows these steps:

• First Component: Subtract the first component of \( \mathbf{u} \) from the first component of \( \mathbf{v} \): \( 4 - (-2) = 4 + 2 = 6 \).
• Second Component: Subtract the second component of \( \mathbf{u} \) from the second component of \( \mathbf{v} \): \( 3 - 5 = -2 \).

Combining these results, we arrive at the resulting vector: \( \mathbf{v} - \mathbf{u} = \langle 6, -2 \rangle \). This process is straightforward once you remember to handle the negative signs carefully, especially when subtracting negative numbers.

The Role of Vector Subtraction in Mathematics Education

Vector subtraction is a foundational topic in mathematics that underpins many concepts in higher education. As students learn to manipulate vectors via operations like addition and subtraction, they build their understanding of both algebraic and geometric perspectives.

• Algebraic Understanding: Teaches students how to apply arithmetic operations to abstract concepts such as vectors, enhancing logical thinking and problem-solving skills.
• Geometric Visualization: Students gain insight into how vectors operate in spaces, developing visual reasoning skills crucial for areas like physics and engineering.
• Cross-disciplinary Significance: Mastering vectors and their operations is fundamental in various fields like physics, computer science, and any domain that deals with dimensions.

Mathematics education emphasizes vector subtraction because it provides a concrete way to explore the relationship between different quantities, fostering a deeper understanding of vector algebra and its applications. This understanding is not only academic but also critical for real-world problem-solving and technology development.