Problem 35 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\left\langle\frac{4}{3}, \frac{8}{15}\right\rangle \text { and }\left\langle-\frac{1}{12}, \frac{5}{24}\right\rangle$$

Step-by-Step Solution

Answer

The vectors are orthogonal.

Step 1: Understand the Problem

To determine if two vectors are orthogonal, you need to check whether their dot product is zero. If the dot product equals zero, then the vectors are orthogonal.

Step 2: Express the Vectors

The vectors given are \( \mathbf{u} = \left\langle\frac{4}{3}, \frac{8}{15}\right\rangle \) and \( \mathbf{v} = \left\langle-\frac{1}{12}, \frac{5}{24}\right\rangle \).

Step 3: Find the Dot Product

The dot product of two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) is calculated as \( u_1 \cdot v_1 + u_2 \cdot v_2 \).

Step 4: Calculate Each Component

Calculate the products of the corresponding components: \( \frac{4}{3} \times -\frac{1}{12} = -\frac{4}{36} = -\frac{1}{9} \).\( \frac{8}{15} \times \frac{5}{24} = \frac{40}{360} = \frac{1}{9} \).

Step 5: Sum the Products

Add the two calculated values: \(-\frac{1}{9} + \frac{1}{9} = 0\).

Step 6: Conclusion

Since the dot product is \(0\), the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It is used to find the result of multiplying two vectors. Simply put, the dot product combines two vectors into a single number (also called a scalar). To calculate the dot product, multiply the corresponding components of each vector and add those products together. For example, if you have vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), their dot product is given by:

• \( u_1 \cdot v_1 + u_2 \cdot v_2 \)

This operation is helpful in many areas of mathematics, physics, and engineering, especially when determining angles between vectors.

Vector Components

Vectors are mathematical objects used to describe quantities with both a magnitude and a direction. In two-dimensional space, a vector is often represented as \( \langle x, y \rangle \), where \( x \) and \( y \) are its components. These components indicate how far the vector stretches along each axis.
For the vectors given in the original exercise, they are defined by their components:

• \( \mathbf{u} = \langle \frac{4}{3}, \frac{8}{15} \rangle \)
• \( \mathbf{v} = \langle -\frac{1}{12}, \frac{5}{24} \rangle \)

Each component is essentially a piece of the vector that points in a direction, which can later be used to compute operations such as the dot product.

Orthogonality Check

An orthogonality check is used to determine if two vectors are perpendicular to each other. Vectors are orthogonal if their dot product is zero. This means they meet at a right angle, effectively conveying no direction-related similarity.
In the original exercise, the vectors \( \mathbf{u} = \langle \frac{4}{3}, \frac{8}{15} \rangle \) and \( \mathbf{v} = \langle -\frac{1}{12}, \frac{5}{24} \rangle \) were analyzed. The dot product was calculated as follows:

• Component multiplication: \( \frac{4}{3} \times -\frac{1}{12} = -\frac{1}{9} \) and \( \frac{8}{15} \times \frac{5}{24} = \frac{1}{9} \)
• Sum of products: \(-\frac{1}{9} + \frac{1}{9} = 0\)

Since the dot product is zero, this means \( \mathbf{u} \) and \( \mathbf{v} \) are indeed orthogonal vectors. This check is a powerful tool in confirming the perpendicularity of vectors in analyses.