Problem 68 - Precalculus with Limits

Question

Finding Orthogonal Vectors In Exercises \(67-70\) , find two vectors in opposite directions that are orthogonal to the vector u. (There are many correct answers.) $$\mathbf{u}=\langle- 8,3\rangle$$

Step-by-Step Solution

Answer

The orthogonal vectors are \( \mathbf{v}=\langle 1, -\frac{8}{3} \rangle \) and \( \mathbf{v}_{opposite}=\langle -1, \frac{8}{3} \rangle \).

Step 1 - Formulate the Orthogonality Condition

The two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal if their dot product equals zero. Given \( \mathbf{u}=\langle- 8,3\rangle \), an orthogonal vector \( \mathbf{v}=\langle x,y \rangle \) should meet the condition -8*x + 3*y = 0.

Step 2 - Construct the Orthogonal Vector

Choose an arbitrary value for one of the variables, then solve for the other one. Let's choose \( x = 1 \), substituting into the equation gives you \( y = -\frac{8}{3} \). So \( \mathbf{v}=\langle 1, -\frac{8}{3} \rangle \) is one solution.

Step 3 - Find the Vector in the Opposite Direction

Multiply \( \mathbf{v} \) by -1. Therefore, \( \mathbf{v}_{opposite}=\langle -1, \frac{8}{3} \rangle \) is the vector in the opposite direction that is also orthogonal to \( \mathbf{u} \).

Key Concepts

Dot Product

In vector mathematics, the dot product is a crucial operation that helps determine properties such as orthogonality. The dot product between two vectors is calculated by multiplying their corresponding components and then summing the results. For example, if we 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:

• \( \mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2 \)

Orthogonality, or perpendicularity, is defined as the dot product being zero. When two vectors are orthogonal, they form a right angle. In simpler terms, they do not influence each other in terms of direction.

Using the example from the exercise where \( \mathbf{u} = \langle -8, 3 \rangle \), if \( \mathbf{v} = \langle x, y \rangle \) is orthogonal to \( \mathbf{u} \), the dot product must satisfy the equation:

• \( -8x + 3y = 0 \)

Vector Operations

Vector operations involve various mathematical procedures on vectors, which include addition, subtraction, scaling, and especially here, finding orthogonal vectors. These operations follow specific rules, enabling complex calculations and solutions in physics and engineering.

To find such orthogonal vectors in the exercise, we started by choosing an arbitrary component for the unknown vector \( \mathbf{v} = \langle x, y \rangle \). By substituting \( x = 1 \) into the orthogonal condition for our vector \( \mathbf{u} = \langle -8, 3 \rangle \), we find:

• \( -8 \times 1 + 3y = 0 \)
• Simplifying gives \( y = -\frac{8}{3} \)

Now, the vector \( \mathbf{v} = \langle 1, -\frac{8}{3} \rangle \) is orthogonal to \( \mathbf{u} \). By further scaling this vector by multiplying each component by \(-1\), we get the opposite direction vector \( \mathbf{v}_{opposite} = \langle -1, \frac{8}{3} \rangle \), which is also orthogonal.

Mathematical Solutions

Finding mathematical solutions to a problem often involves strategic choices and a methodical approach. In this case, solving for orthogonal vectors in opposite directions required setting up a system based on the orthogonality criteria, as highlighted by the dot product.

First, identify what the condition or property needed is, such as orthogonality. Next, translate this need into a mathematical expression or equation, like

• \( -8x + 3y = 0 \)

This represents all vectors \( \mathbf{v} = \langle x, y \rangle \) orthogonal to \( \mathbf{u} = \langle -8, 3 \rangle \).

Afterward, an arbitrary selection of one variable leads to solving for the other. Here, selecting \( x = 1 \) led to finding \( y = -\frac{8}{3} \). This simple strategic choice simplifies solving these problems. By employing these steps, and using multiplication for opposite vectors, solutions become straightforward and highlight the power of mathematical strategies in reaching precise answers.