Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Determine whether each pair of vectors is orthogonal. $$\langle 1,1\rangle,\langle 1,-1\rangle$$

Step-by-Step Solution

Answer

The vectors \( \langle 1, 1 \rangle \) and \( \langle 1, -1 \rangle \) are orthogonal.

Step 1: Understand Orthogonal Vectors

Two vectors are orthogonal if their dot product is equal to zero. Let's consider the vectors \( \langle 1, 1 \rangle \) and \( \langle 1, -1 \rangle \).

Step 2: Calculate the Dot Product

The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is given by \( a \times c + b \times d \). For the vectors \( \langle 1, 1 \rangle \) and \( \langle 1, -1 \rangle \), we calculate: \[1 \times 1 + 1 \times (-1) = 1 - 1 = 0.\]

Step 3: Determine Orthogonality Based on Dot Product

Since the dot product is 0, the vectors \( \langle 1, 1 \rangle \) and \( \langle 1, -1 \rangle \) are orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental operation in linear algebra when dealing with vectors. It provides a way to measure the cosine of the angle between two vectors, which helps us determine their relationship. The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is calculated by multiplying the corresponding components and summing the results: \[ a \times c + b \times d. \]

• If the dot product is positive, the angle between the vectors is less than 90 degrees.
• If the dot product is zero, the vectors are orthogonal, meaning they are perpendicular to each other at a right angle.
• If the dot product is negative, the angle is more than 90 degrees.

Here, we found the dot product of the vectors \( \langle 1, 1 \rangle \) and \( \langle 1, -1 \rangle \) to be zero, indicating the vectors are orthogonal. This is a handy property for determining vector relationships, particularly in physics and computer graphics.

Vector Analysis

Vector analysis involves examining vectors to understand their characteristics and relationships. A vector is characterized by both a magnitude (length) and a direction. When analyzing vectors, one often uses operations such as addition, subtraction, multiplication (dot product and cross product), and others. In our exercise, vectors \( \langle 1, 1 \rangle \) and \( \langle 1, -1 \rangle \) were examined for orthogonality. The tools of vector analysis, such as the dot product, help us understand how vectors interact in multidimensional space.Some key points in vector analysis include:

• Magnitude: The length of the vector, calculated using Euclidean distance.
• Direction: Given by the vector’s angle concerning a reference axis.
• Orthogonality: Indicates perpendicular vectors, which the dot product can easily identify.

Understanding these concepts allows for the manipulation and application of vectors in engineering, physics, and computer science.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, matrices, and linear transformations. It’s a powerful tool used across various scientific fields, including computer science, economics, and engineering. Vectors and operations on vectors, like the dot product, are central concepts in linear algebra.

• Linear Equations: Linear algebra often deals with systems of linear equations and solutions.
• Matrix Operations: Matrices, which can represent systems of equations or transformations, are manipulated similarly to vectors.
• Vector Spaces: Collections of vectors that can be added and multiplied by scalars to produce another vector.
• Orthogonality: Within vector spaces, linear algebra provides the tools to determine perpendicularity, or orthogonality, which is crucial for simplifying problems and understanding vector behaviour.

In our task, understanding the orthogonality of vectors is an application of linear algebra and offers insight into how these vectors interact systematically. Recognizing perpendicularity in vectors can simplify problem-solving and lead to better, more efficient solutions in practical applications.