Question

Consider the vectors \(\mathbf{u}=\langle\cos \alpha, \sin \alpha, 0\rangle\) and \(\mathbf{v}=\langle\cos \beta, \sin \beta, 0\rangle\) where \(\alpha>\beta\) Find the dot product of the vectors and use the result to prove the identity \(\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\).

Short answer

The dot product of the vectors is \(\cos\alpha\cos\beta + \sin\alpha\sin\beta\). Substituting this into the dot product formula, we have proved the identity \(\cos (\alpha-\beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\).

Step-by-step solution

Compute the Dot Product

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by \(\mathbf{u}\cdot\mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos\theta\), where \(\theta\) is the angle between the two vectors. So we first compute the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) which is \(\mathbf{u}\cdot\mathbf{v} = \cos\alpha\cos\beta + \sin\alpha\sin\beta + 0x0 = \cos\alpha\cos\beta + \sin\alpha\sin\beta\)

Find the Angle Between the Vectors

The angle between the vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \(\alpha - \beta\). We plug this into the dot product formula \(\mathbf{u}\cdot\mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos\theta\), and since the lengths of \(\mathbf{u}\) and \(\mathbf{v}\) are 1 (as they are unit vectors) we have \(\cos\alpha\cos\beta + \sin\alpha\sin\beta = \cos(\alpha-\beta)\)

Confirm the Identity

Now, we can see that we indeed have the trigonometric identity \(\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\), which is what we had set out to prove.

Key concepts

Vectors in Calculus

Vectors are fundamental in calculus, especially when handling multi-dimensional problems. A vector, often denoted with a bold letter such as \textbf{v}, is a quantity with both magnitude and direction. In the context of the given exercise, the vectors \textbf{u} and \textbf{v} are defined in two dimensions, and their components are functions of the angles \(\alpha\) and \(\beta\), respectively.

Vectors are often represented in a coordinate system with components that can take the form of trigonometric functions, as is the case here. Calculating the dot product, or scalar product, of two vectors involves multiplying corresponding components of two vectors and summing up the products. In two dimensions, this is \(\mathbf{u_1}\cdot\mathbf{v_1} + \mathbf{u_2}\cdot\mathbf{v_2}\), and in three dimensions, an additional term is added for the third component.

\textbf{u} and \textbf{v} in the original problem are simplified to not include the third dimension because their third components are zero. This demonstrates a fundamental property of vectors: any component that is zero means that the vectors have no magnitude in that direction.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions of angles to one another. They are immensely useful in simplifying expressions in geometry, calculus, and beyond. One of the foundational identities that students encounter is the cosine of a difference between two angles, \(\cos(\alpha-\beta)\), which can be expressed as the sum of products of cosines and sines of these angles.

Stemming from the exercise discussed, knowing the identity \(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\) allows us to use the dot product of vectors to reveal this relationship between angles. The step-by-step solution provided gives us a concrete example of how trigonometric identities can be visually and analytically derived from geometric concepts like vector dot products.

Understanding these identities is not only about memorizing equations; it's also about recognizing patterns and connections within different branches of mathematics such as algebra and geometry, making problem-solving more intuitive.

Angle Between Vectors

The angle between two vectors is a measure of how they are oriented with respect to each other. It's an integral concept in vector calculus. Knowing how to find this angle is essential in many applications, including physics, computer graphics, and navigation.

In the context of the dot product, the angle can be found using the formula cited in the step-by-step solution: \(\mathbf{u}\cdot\mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos\theta\), where \(\theta\) is the angle we want to determine. Here, \(\mathbf{u}\cdot\mathbf{v}\) is the dot product, and \(|\mathbf{u}|\) and \(|\mathbf{v}|\) are the magnitudes of vectors \(\mathbf{u}\) and \(\mathbf{v}\), respectively. The vectors in the problem have been specifically given in a way that corresponds to the unit circle, which makes their magnitudes 1.

Real-world applications often require calculating angles between vectors to determine relative direction. As such, visualizing vectors as arrows from the origin to a point in space can also help in understanding how angles between vectors are an expression of their directional relationship.