Problem 70 - 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.) $$\mathbf{u}=-\frac{5}{2} \mathbf{i}-3 \mathbf{j}$$

Step-by-Step Solution

Answer

The two vectors that are orthogonal to vector \(\mathbf{u} = -\frac{5}{2} \mathbf{i} - 3 \mathbf{j}\) are \(v = \mathbf{i}-\frac{5}{6}\mathbf{j}\) and \(v' = \mathbf{i}+\frac{5}{6}\mathbf{j}\)

Step 1: Determine x and y

Given that the vectors \(u\) = \(-\frac{5}{2}\mathbf{i}-3 \mathbf{j}\) and \(v = x\mathbf{i}+y\mathbf{j}\) are orthogonal, we have the condition for orthogonality as \(ax+by = 0\) or \(-\frac{5}{2}x-3y = 0\). Let's take x = 1, then solving for y we get \(y = -\frac{-5x}{6} = -\frac{5}{6}\). Therefore one vector \(v = \mathbf{i}-\frac{5}{6}\mathbf{j}\)

Step 2: Find the vector in the opposite direction

A vector in opposite direction can be obtained by simply changing the sign of either x or y. Let's change the sign of y, then we get \(v' = \mathbf{i}+\frac{5}{6}\mathbf{j}\). Thus \(v'\) is the vector in the opposite direction

Step 3: Confirmation

As a check, you can confirm these results by taking the dot product of vector \(u\) with vectors \(v\) and \(v'\) respectively. The dot product should be zero in each case because orthogonal vectors have a dot product of zero

Key Concepts

Vector Orthogonality

In the realm of geometry and precalculus, understanding vector orthogonality is crucial for comprehending how vectors interact in space. Two vectors are considered orthogonal if they meet at a right angle, which is equivalent to a 90-degree intersection.

When dealing with vectors in the xy-plane, orthogonality can be visually imagined as two arrows crossing each other perpendicularly. In the given exercise, you are asked to find vectors that are orthogonal to a specific vector, \( \mathbf{u} \). This means you need to find vectors that would cross \( \mathbf{u} \) exactly at a right angle.

Using a coordinate system, the concept of orthogonality is often verified algebraically by employing the dot product. If the dot product of two vectors is zero, this quickly tells us that the vectors are orthogonal, without the need for visual confirmation. Orthogonal vectors play a key role in various applications, from computer graphics to physics, emphasizing the importance of mastering this concept in precalculus studies.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single number, a scalar. It's a fundamental concept not only in precalculus but also in higher-level mathematics and physics.

To find the dot product of two vectors \( \mathbf{a} = a_{x}\mathbf{i} + a_{y}\mathbf{j} \) and \( \mathbf{b} = b_{x}\mathbf{i} + b_{y}\mathbf{j} \), we multiply their corresponding components and add the results: \( \mathbf{a} \cdot \mathbf{b} = a_{x}b_{x} + a_{y}b_{y} \).

The beauty of the dot product lies in its ability to reveal the relationship between two vectors. If the result is zero, it means the vectors are orthogonal. This property provides an efficient way to find vectors that are orthogonal to a given vector, just like the ones you are asked to find in the exercise. By setting up an equation based on the dot product and solving for the variable components, you can determine vectors that satisfy the orthogonal condition.

Precalculus

The field of precalculus serves as the bedrock for understanding calculus, and it involves studying various topics such as functions, equations, and, as covered here, vectors. Precalculus aims to equip students with the mathematical tools and thinking necessary to handle the more complex concepts they will encounter in calculus.

In relation to vectors, precalculus covers the concept of vector operations, including addition, subtraction, and scalar multiplication, and also dives into vector properties like magnitude, direction, and the aforementioned orthogonality and dot product. Problems such as finding orthogonal vectors demonstrate the practical application of precalculus principles to geometric interpretations.

While precalculus can be challenging, grasping these fundamentals is imperative for success in higher-level math and various scientific fields. Clear understanding of vector operations and properties provides a strong basis for not only calculus but also for real-world problem-solving in engineering, physics, and computer science.