Problem 71 - Precalculus with Calculus Previews

Question

Determine whether the given vectors are orthogonal. $$ 4 \mathbf{i}-5 \mathbf{j}, \mathbf{i}+\frac{4}{5} \mathbf{j} $$

Step-by-Step Solution

Answer

The vectors are orthogonal because their dot product is zero.

Step 1: Understand Orthogonal Vectors

Two vectors are orthogonal if their dot product is zero. Let's denote the vectors as \( \mathbf{a} = 4\mathbf{i} - 5\mathbf{j} \) and \( \mathbf{b} = \mathbf{i} + \frac{4}{5} \mathbf{j} \). We need to compute \( \mathbf{a} \cdot \mathbf{b} \).

Step 2: Calculate the Dot Product

The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as the sum of the products of their corresponding components. So, \( \mathbf{a} \cdot \mathbf{b} = (4)(1) + (-5)\left(\frac{4}{5}\right) \).

Step 3: Simplify the Dot Product

Calculate each product: \( 4 \times 1 = 4 \) and \( -5 \times \frac{4}{5} = -4 \). Now add these results: \( 4 + (-4) = 0 \).

Step 4: Conclusion on Orthogonality

Since the dot product is zero, \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal vectors.

Key Concepts

Dot Product

In vector mathematics, the dot product is a fundamental operation that helps us understand relationships between two vectors. The dot product, also known as the scalar product, is calculated as the sum of the products of the corresponding components from each vector. The result of a dot product is a single number (a scalar), not a vector. Consider two vectors, \( \mathbf{a} = 4\mathbf{i} - 5\mathbf{j} \) and \( \mathbf{b} = \mathbf{i} + \frac{4}{5} \mathbf{j} \).
The dot product is calculated as:

• Multiply the x-components: \( 4 \times 1 \).
• Multiply the y-components: \( -5 \times \frac{4}{5} \).
• Add these products together: \( 4 + (-4) \).

The result is \( 0 \), which indicates that the vectors are orthogonal (perpendicular). Dot products are particularly useful in physics and engineering to determine if vectors are perpendicular, simplify angle calculations, and project one vector onto another.

Vector Mathematics

Vector mathematics is a crucial part of understanding and solving problems in physics, engineering, and computer graphics. Vectors have both magnitude and direction, which makes them ideal to describe physical quantities like velocity, force, and displacement. A vector is typically represented in the form \( a\mathbf{i} + b\mathbf{j} \) in two-dimensional space, where \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors along the x and y axes, respectively.
Let's break down the vector equation \( \mathbf{a} = 4\mathbf{i} - 5\mathbf{j} \):

• The coefficient \( 4 \) is the component in the x-direction, and \( -5 \) is in the y-direction.
• This signifies that vector \( \mathbf{a} \) moves 4 units in the positive x-direction and 5 units in the negative y-direction.

Understanding vectors is essential for learning more advanced concepts, such as vector addition, subtraction, and the dot product, which deepen our ability to analyze and solve multi-directional problems accurately.

Analytical Geometry

Analytical geometry, also known as coordinate geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. Analytical geometry allows us to model geometric problems algebraically, making them easier to solve. When dealing with vectors like \( \mathbf{i} \) and \( \mathbf{j} \), we use a coordinate plane to describe the direction and magnitude of the vectors.
In this context, the goal was to verify if two vectors were orthogonal by calculating their dot product. In analytical geometry:

• Vectors are orthogonal if their dot product is zero.
• The coordinate plane helps visualize vector orientation and operations like addition or scalar multiplication.

This realm of geometry is crucial for understanding spatial relationships and transformations, aiding fields like computer graphics by defining rotations, translations, and transformations mathematically.