Problem 69 - 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.) $$\mathbf{u}=\frac{1}{2} \mathbf{i}-\frac{2}{3} \mathbf{j}$$

Step-by-Step Solution

Answer

The vectors orthogonal to \(\mathbf{u}=\frac{1}{2}\mathbf{i}-\frac{2}{3}\mathbf{j}\) are \(\mathbf{v}_1=2\mathbf{i}+\frac{3}{2}\mathbf{j}\) and \(-\mathbf{v}_1 = -2\mathbf{i} - \frac{3}{2}\mathbf{j}\).

Step 1: Understanding Orthogonality

In 2D geometry, two vectors are orthogonal if their dot product is zero. The dot product of vectors \(\mathbf{u}=\frac{1}{2}\mathbf{i}-\frac{2}{3}\mathbf{j}\) and a vector \(\mathbf{v}=a\mathbf{i}+b\mathbf{j}\) is given by: \( \mathbf{u} \cdot \mathbf{v} = a\frac{1}{2} + b(-\frac{2}{3}) = 0\). This equation will help us find the values of a and b, which define vector \(\mathbf{v}\) that is orthogonal to \(\mathbf{u}\).

Step 2: Finding Vector One

Choose a value for \(a\) or \(b\). Let's choose \(a = 2\). Substitute \(a\) into the equation from Step 1 to find \(b\): \(2\frac{1}{2} + b(-\frac{2}{3}) = 0\). Solving this gives \(b = \frac{3}{2}\). Therefore, one vector \(\mathbf{v}_1\) that is orthogonal to \(\mathbf{u}\) is \(\mathbf{v}_1=2\mathbf{i}+\frac{3}{2}\mathbf{j}\).

Step 3: Finding Vector Two

To find an opposite direction vector, simply change the sign of the first vector. Therefore, another vector that is orthogonal to \(\mathbf{u}\) is \(-\mathbf{v}_1 = -2\mathbf{i} - \frac{3}{2}\mathbf{j}\).

Key Concepts

2D Geometry

In the world of 2D geometry, we focus on objects and problems that exist on a flat plane, like a piece of paper. Vectors are key components in this plane because they represent directions and magnitudes. These directions are crucial in calculations and in analyzing geometric problems. By understanding how vectors behave, such as determining if they are orthogonal, we can solve a wide range of mathematical and real-world problems. This allows us to predict movements, organize patterns, and manipulate two-dimensional objects in various fields, from computer graphics to engineering.

Dot Product

The dot product is a mathematical operation that helps us find important relationships between vectors. Specifically, in 2D geometry, it shows us whether two vectors are orthogonal. The dot product is calculated by multiplying corresponding components of two vectors and summing the results. Given vectors \( \mathbf{u} = \frac{1}{2} \mathbf{i} - \frac{2}{3} \mathbf{j} \) and \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \), the dot product is \( \mathbf{u} \cdot \mathbf{v} = a\frac{1}{2} + b(-\frac{2}{3}) \).
This operation results in a single number called a scalar, and when it equals zero, the vectors are orthogonal. This indicates a 90-degree angle between them. Thus, the dot product is a powerful tool for analyzing angles and relationships between vectors.

Vector u

Let's take a closer look at vector \( \mathbf{u} \), which is given in the problem as \( \mathbf{u} = \frac{1}{2}\mathbf{i} - \frac{2}{3}\mathbf{j} \). It consists of two parts: \( \mathbf{i} \), the horizontal unit vector, and \( \mathbf{j} \), the vertical unit vector, with coefficients \( \frac{1}{2} \) and \(-\frac{2}{3} \) respectively. Vectors like \( \mathbf{u} \) hold both direction and magnitude, playing a crucial role in calculations that require both these components.
This particular vector moves halfway in the positive \( x \)-direction and more than halfway in the negative \( y \)-direction. Understanding these components allows us to easily construct vectors that fit specific requirements, such as being orthogonal, and find compounds of movements within a 2D space.

Vector Direction

Vectors have both magnitude and direction. The direction is what distinguishes vectors from mere numbers, and it's represented by the vector's components. When looking for vectors orthogonal to \( \mathbf{u} \), we're concerned with their directions. If two vectors are orthogonal, their directions form a 90-degree angle or they lie perpendicular to each other in a 2D plane.
When solving our original problem, finding multiple orthogonal vectors involves choosing distinct directions while ensuring one is the negative of the other. This is done by inverting the signs of all components to yield an opposite vector. Thus, direction dictates not only the vector's path but also how it interacts geometrically with another vector, especially in terms of orthogonality.