Problem 41 - Algebra and Trigonometry

Question

Suppose that \(\mathbf{v}\) and \(\mathbf{w}\) are unit vectors. If the angle between \(\mathbf{v}\) and \(\mathbf{i}\) is \(\alpha\) and the angle between \(\mathbf{w}\) and \(\mathbf{i}\) is \(\beta\), use the idea of the dot product \(\mathbf{v} \cdot \mathbf{w}\) to prove that $$ \cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta $$

Step-by-Step Solution

Answer

\( \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \)

Step 1 - Recall the definition of the dot product

The dot product of two vectors \(\textbf{v}\) and \(\textbf{w}\) is defined as \(\textbf{v} \cdot \textbf{w} = |\textbf{v}| |\textbf{w}| \cos \theta\), where \(\theta\) is the angle between the vectors \(\textbf{v}\) and \(\textbf{w}\).

Step 2 - Use the fact that \(\textbf{v}\) and \(\textbf{w}\) are unit vectors

Since \(\textbf{v}\) and \(\textbf{w}\) are unit vectors, we have \(|\textbf{v}| = 1\) and \(|\textbf{w}| = 1\). Therefore, the dot product simplifies to \(\textbf{v} \cdot \textbf{w} = \cos \theta\).

Step 3 - Express vectors \(\textbf{v}\) and \(\textbf{w}\) in terms of their angles with \(\textbf{i}\)

Given that the angle between \(\textbf{v}\) and \(\textbf{i}\) is \(\textbf{α}\) and the angle between \(\textbf{w}\) and \(\textbf{i}\) is \(\textbf{β}\), we can write \(\textbf{v}\) and \(\textbf{w}\) in terms of their components: \(\textbf{v} = \cosα \textbf{i} + \sinα \textbf{j}\) and \(\textbf{w} = \cosβ \textbf{i} + \sinβ \textbf{j}\).

Step 4 - Compute the dot product \(\textbf{v} \cdot \textbf{w}\)

Using the component form of the vectors, the dot product \(\textbf{v} \cdot \textbf{w}\) becomes: \((\cosα \textbf{i} + \sinα \textbf{j}) \cdot (\cosβ \textbf{i} + \sinβ \textbf{j}) \). Expanding this, we get: \(\textbf{v} \cdot \textbf{w} = \cosα \cos β + \sinα \sinβ\).

Step 5 - Connect the dot product and the cosine of the angle

From Steps 1 and 2, we know that \(\textbf{v} \cdot \textbf{w} = \cos(\theta)\), where \(\theta\) is the angle between \(\textbf{v}\) and \(\textbf{w}\). Therefore, \(\textbf{v} \cdot \textbf{w} = \cos (\textbf{α} - \textbf{β})\).

Step 6 - Finalize the equation

Since we have \(\textbf{v} \cdot \textbf{w} = \cos(\textbf{α} - \textbf{β})\) and from Step 4, we have \(\textbf{v} \cdot \textbf{w} = \cos α \cos β + \sin α \sin β\), we can equate them to get the desired identity: \(\textbf{cos} (\textbf{α} - \textbf{β}) = \cos α \cos β + \sin α \sin β\).

Key Concepts

unit vectors

When discussing vectors in mathematics and physics, unit vectors are particularly important. A unit vector is a vector with a length (or magnitude) of 1. This makes them useful for indicating directions without affecting magnitudes.

• Notation: Unit vectors are usually denoted with a hat, for example, \(\textbf{\hat{i}}\) and \(\textbf{\hat{j}}\).
• Properties: For any vector \(\textbf{v}\), dividing it by its magnitude \(|\textbf{v}|\) converts it into a unit vector, represented as \(\textbf{\hat{v}} = \frac{\textbf{v}}{|\textbf{v}|}\).
• Application: In the given problem, both \(\textbf{v}\) and \(\textbf{w}\) are unit vectors, meaning their magnitudes are 1. This simplifies calculations and the use of dot product.

dot product

The dot product (also known as scalar product) is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It's especially useful in projections and angle calculations between vectors.

• Definition: For vectors \(\textbf{v}\) and \(\textbf{w}\), the dot product is given by \(\textbf{v} \cdot \textbf{w} = |\textbf{v}| |\textbf{w}| \cos \theta\), where \(\theta\) is the angle between the vectors.
• Simplification for unit vectors: Since \(|\textbf{v}| = 1\) and \(|\textbf{w}| = 1\) for unit vectors, this becomes \(\textbf{v} \cdot \textbf{w} = \cos \theta\).
• Use in the exercise: The dot product allows us to express the angle \(\theta\) between vectors \(\textbf{v}\) and \(\textbf{w}\) in terms of their components, aiding in the derivation of the trigonometric identity.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are key tools in simplifying and solving trigonometric expressions and equations.

• Basic identities: Common trigonometric identities include \(\sin^2\alpha + \cos^2\alpha = 1\) and \(\tan\alpha = \frac{\sin \alpha}{\cos \alpha}\).
• Specific to the exercise: The cosine angle difference identity, \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\), helps in finding the cosine of the difference between two angles. It’s derived from the properties of dot product and components of unit vectors.
• Utility: These identities enable the conversion of complex trigonometric expressions into simpler forms, often turning multiplications into additions or subtractions.

cosine and sine functions

The cosine and sine functions are fundamental in trigonometry, representing the coordinates of a point on the unit circle. Understanding these functions is crucial for working with trigonometric identities and solving related problems.

• Definition: \(\cos \alpha\) represents the x-coordinate and \(\sin \alpha\) represents the y-coordinate of a point on the unit circle at an angle \(\alpha\) from the positive x-axis.
• Relationship: They are periodic functions with a period of \(2\pi\), meaning \(\cos(\alpha + 2\pi) = \cos \alpha\) and \(\sin(\alpha + 2\pi) = \sin \alpha\).
• In the problem: By expressing unit vectors in terms of cosine and sine (e.g., \(\textbf{v} = \cos \alpha \textbf{i} + \sin \alpha \textbf{j}\) and \(\textbf{w} = \cos \beta \textbf{i} + \sin \beta \textbf{j}\)), we can effectively use these functions to prove the given trigonometric identity.