Problem 54 - Precalculus with Limits

Question

Determining Orthogonal Vectors In Exercises \(53-58\) , determine whether \(u\) and \(v\) are orthogonal. $$\mathbf{u}=\langle 3,15\rangle$$ $$\mathbf{v}=\langle- 1,5\rangle$$

Step-by-Step Solution

Answer

The vectors \( \mathbf{u} = \langle 3, 15 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \) are not orthogonal.

Step 1: Recall the definition of orthogonal vectors

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal (or perpendicular) if and only if their dot product is zero. The dot product \(\mathbf{u} \cdot \mathbf{v}\) is calculated as \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2\), where \(u_1, u_2\) are the components of \(\mathbf{u}\) and \(v_1, v_2\) are the components of \(\mathbf{v}\).

Step 2: Calculate the dot product

Use the given vectors \(\mathbf{u} = \langle 3, 15 \rangle\) and \(\mathbf{v} = \langle -1, 5 \rangle\) to calculate the dot product. \(\mathbf{u} \cdot \mathbf{v} = (3*(-1)) + (15*5) = -3 + 75 = 72.

Step 3: Determine Orthogonality

As the dot product \(\mathbf{u} \cdot \mathbf{v} = 72\) is not equal to zero, vectors \(\mathbf{u}\) and \(\mathbf{v}\) are not orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics that helps us understand the relationship between two vectors. Given two vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product \( \mathbf{u} \cdot \mathbf{v} \) is calculated as follows:

• Multiply each pair of corresponding components from the two vectors.
• Add the results of these multiplications together.

For example, if \( \mathbf{u} = \langle 3, 15 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \), the dot product is computed as:\[ \mathbf{u} \cdot \mathbf{v} = (3 \cdot -1) + (15 \cdot 5) = -3 + 75 = 72 \]A critical property of the dot product is that it reflects the angle between the vectors. Specifically, the dot product is zero if the vectors are orthogonal (at a right angle to each other). Understanding how to compute and interpret the dot product is crucial for determining if two vectors are perpendicular.

Vector Components

Vectors are mathematical objects characterized by both magnitude and direction. Each vector can be broken down into components that represent its projection along each axis in a given coordinate system. For example, the vector \( \mathbf{u} = \langle 3, 15 \rangle \) consists of two components:

• \( 3 \) is the component along the x-axis.
• \( 15 \) is the component along the y-axis.

These components are essential for performing operations like the dot product or finding the magnitude and direction of a vector.The components of a vector allow us to easily perform calculations and apply physics principles. When you have two vectors, their components can be used to explore how vectors interact with each other, such as determining if they are perpendicular by using the dot product.

Perpendicular Vectors

Perpendicular, or orthogonal, vectors have a special geometric relationship: they meet at a right angle, or 90 degrees. This concept is essential in understanding spatial relationships in vector spaces. Two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular if their dot product is zero:\[ \mathbf{u} \cdot \mathbf{v} = 0 \]To check if vectors are perpendicular, calculate their dot product using their components. If the result is zero, the vectors are orthogonal. For instance, with \( \mathbf{u} = \langle 3, 15 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \), the dot product is 72, which indicates they are not perpendicular.The concept of perpendicular vectors is widely applicable, from physics to engineering, as it helps in analyzing forces, balancing systems, and designing structures.