Problem 53 - Precalculus with Limits

Question

Determining Orthogonal Vectors In Exercises \(53-58\) , determine whether \(u\) and \(v\) are orthogonal. $$\begin{aligned} \mathbf{u} &=\langle- 12,30\rangle \\ \mathbf{v} &=\left\langle\frac{1}{2},-\frac{5}{4}\right\rangle \end{aligned}$$

Step-by-Step Solution

Answer

The vectors \( \mathbf{u} = \langle -12, 30 \rangle \) and \( \mathbf{v} = \langle \frac{1}{2}, -\frac{5}{4} \rangle \) are not orthogonal.

Step 1: Identify the components of vectors u and v

The components of vector \( \mathbf{u} = \langle -12, 30 \rangle \) are \( u_1 = -12 \) and \( u_2 = 30 \); similarly, the components of vector \( \mathbf{v} = \langle \frac{1}{2}, -\frac{5}{4} \rangle \) are \( v_1 = \frac{1}{2} \), and \( v_2 = -\frac{5}{4} \).

Step 2: Calculate the dot product

The dot product \( \mathbf{u} \cdot \mathbf{v} \) is computed as follows: \( (-12) \cdot (\frac{1}{2}) + (30) \cdot (-\frac{5}{4}) = -6 - \frac{150}{4} = -6 - 37.5 = -43.5.

Step 3: Determine if u and v are orthogonal

Vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal if their dot product is 0. However, as per the calculation in Step 2, \( \mathbf{u} \cdot \mathbf{v} = -43.5 \), which is not equal to 0. Hence, vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It's a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. Calculating the dot product of two vectors can reveal a lot about their relationship. To find the dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \), use the formula:

• Multiply the corresponding components of each vector
• Add those products together

For example, if \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), then their dot product \( \mathbf{u} \cdot \mathbf{v} \) is \( (u_1 \cdot v_1) + (u_2 \cdot v_2) \). The result is a scalar, not a vector.
This scalar value can help in understanding the angle between the two vectors. Specifically, if the dot product is zero, the vectors are perpendicular or orthogonal to each other.

Vector Components

Vectors are quantities that possess both magnitude and direction. The simplest type of vectors you will frequently encounter are those in two and three dimensions. These can be thought of as arrows pointing in space. The individual numbers making up these vectors are called their components.
For example, consider vector \( \mathbf{u} = \langle -12, 30 \rangle \). Here, \(-12\) is the horizontal component, and \(30\) is the vertical component. Each component represents the influence of the vector in its respective direction. Knowing the components of a vector is crucial for many operations, including finding the dot product.

• Horizontal component is along the x-axis
• Vertical component is along the y-axis

This simplicity in breaking down vectors into components makes complex vector calculations more manageable, as each dimension can be dealt with individually.

Calculating Orthogonality

To determine if two vectors are orthogonal, you simply need to calculate their dot product. Such a calculation involves applying the operations mentioned above for computing the dot product of the given vectors. Once the dot product is obtained, you can then make the determination:

• If the dot product is zero, the vectors are orthogonal (perpendicular to each other)
• If the dot product is not zero, the vectors are not orthogonal

In the problem, vectors \( \mathbf{u} = \langle -12, 30 \rangle \) and \( \mathbf{v} = \langle \frac{1}{2}, -\frac{5}{4} \rangle \) have a dot product of \(-43.5\), indicating that they are not orthogonal. Checking the orthogonality helps in understanding vector relationships such as balance and equilibrium in physical systems.

Mathematical Proof

Mathematical proof involves logical deduction and verification of a mathematical statement. For determining orthogonality, the proof centers on the calculation method structured through the dot product.

• Begin by identifying the vector components
• Next, compute the dot product from these components
• Finally, evaluate the result

If the result is zero, as proven, the vectors are orthogonal. Otherwise, they are not. Having this methodology creates clear and definitive answers about vector relationships. In this exercise, the dot product of \(-43.5\) irrefutably proves the vectors are not orthogonal, as orthogonality strictly requires a dot product of zero. Thus, the calculation serves as a simple yet powerful proof of orthogonality (or the lack thereof).