Problem 58 - Precalculus with Limits

Question

Determining Orthogonal Vectors In Exercises \(53-58\) , determine whether \(u\) and \(v\) are orthogonal. $$\begin{aligned} \mathbf{u} &=\langle\cos \theta, \sin \theta\rangle \\\ \mathbf{v} &=\langle\sin \theta,-\cos \theta\rangle \end{aligned}$$

Step-by-Step Solution

Answer

Yes, the vectors \(u = <\cos \theta, \sin \theta>\) and \(v = <\sin \theta, -\cos \theta>\) are orthogonal to each other.

Step 1: Understand the dot product

The dot product or scalar product of two vectors can be calculated by summing the products of their corresponding elements. Therefore, the dot product of \(u\) and \(v\) vectors is given by \(u \cdot v = u_1v_1 + u_2v_2\) where \(u_1, u_2\) are the components of vector \(u\) and \(v_1, v_2\) are the components of vector \(v\).

Step 2: Substitution for the dot Product

Next we calculate the dot product of vectors \(u\) and \(v\) by substituting the given values for the elements of \(u\) and \(v\) into the dot product formula. \(u \cdot v = (\cos \theta)(\sin \theta) + (\sin \theta)(-\cos \theta)\).

Step 3: Calculation & check for Orthogonality

When we simplify the above expression, we get \(u \cdot v = \cos \theta \sin \theta - \cos \theta \sin \theta = 0\). Since the dot product of vectors \(u\) and \(v\) is zero, they are orthogonal to each other.

Key Concepts

Dot Product

When we talk about the dot product, also known as the scalar product, we're referring to a way to multiply two vectors together to get a scalar (a single number), unlike the cross product, which returns a vector. Mathematicians and physicists often use this concept when they need to find the angle between two vectors or determine whether the vectors are orthogonal.

The formula for the dot product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\), with respective components \(a_1, a_2, ..., a_n\) and \(b_1, b_2, ..., b_n\), is \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + ... + a_nb_n\).

To understand this better, visualize two arrows pointing in different directions. The dot product tells you how much one vector extends in the direction of the other. It's the 'amount' of one vector that goes in the direction of the second vector. When the dot product is zero, it indicates that the vectors are perpendicular—or orthogonal—to each other, meaning they meet at a right angle.

This concept is crucial not just in mathematical theory but also in various applications like computer graphics, physics, and engineering, where understanding the relationship between different directions and forces is essential.

Vector Orthogonality

Vector orthogonality is a key concept in geometry and vector algebra describing a relationship between two vectors. If two vectors are orthogonal, they are perpendicular to each other at a right angle (90 degrees). In terms of movement or force, two orthogonal vectors do not influence each other; they're independent directions.

Consider a room's corner, where the walls meet the floor. The wall is 'orthogonal' to the floor because they meet at a right angle. Similarly, in vector terms, if two vectors are orthogonal, you can think of them like the edges of a box, meeting at a corner, independent of each other.

To determine vector orthogonality mathematically, we use the dot product. If \(\mathbf{a} \cdot \mathbf{b} = 0\), then the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal. This property makes it easier to solve problems related to projections, optimizations, and geometric interpretations.

Scalar Product

The scalar product, synonymously known as the dot product, refers to an operation that combines two vectors, resulting in a scalar. This scalar is a quantity described by magnitude only, without direction—contrary to vectors, which have both magnitude and direction.

The computation is straightforward: simultaneously multiply the corresponding components of two vectors and then add those products together. For two-dimensional vectors \(\mathbf{a} = (a_x, a_y)\) and \(\mathbf{b} = (b_x, b_y)\), the scalar product is \(\mathbf{a} \cdot \mathbf{b} = a_xb_x + a_yb_y\).

It's helpful to remember that through the scalar product, we can deduce valuable information, such as the angle between two vectors. If two vectors are orthogonal, their scalar product will be zero. This gives us a tool to analyze and solve various problems in physics, such as work done by a force (where force and displacement are vectors), emphasizing its cross-disciplinary relevance.