Problem 17 - Classical Dynamics of Particles and Systems

Question

Obtain the cosine law of plane trigonometry by interpreting the product \((\mathbf{A}-\mathbf{B})\) \((\mathbf{A}-\mathbf{B})\) and the expansion of the product.

Step-by-Step Solution

Answer

To obtain the cosine law of plane trigonometry through the given vector product, we followed these steps: 1. Represented the triangle vertices with vectors \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\), and denoted the opposite sides as a, b, and c. 2. Expanded the dot product \((\mathbf{A}-\mathbf{B})\cdot(\mathbf{A}-\mathbf{B})\), resulting in \(\mathbf{A}\cdot\mathbf{A} - \mathbf{A}\cdot\mathbf{B} - \mathbf{B}\cdot\mathbf{A} + \mathbf{B}\cdot\mathbf{B}\). 3. Simplified the expression to \(a^2 - 2(\mathbf{A}\cdot\mathbf{B}) + b^2\). 4. Substituted the dot product definition, \(\mathbf{A}\cdot\mathbf{B} = ab\cos{\theta}\), where \(\theta\) is the angle between \(\mathbf{A}\) and \(\mathbf{B}\). 5. Obtained the cosine law: \(c^2 = a^2 - 2(ab\cos{\theta}) + b^2\).

Step 1: Write down the vector form of the triangle

Let the vectors \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) represent the three vertices of a triangle. Let the sides opposite to vertices A, B, and C be denoted by a, b, and c respectively.

Step 2: Expand the product of vectors

We want to expand the product \((\mathbf{A}-\mathbf{B})\) \((\mathbf{A}-\mathbf{B})\). Since it represents a dot product, let's write it as \((\mathbf{A}-\mathbf{B})\cdot(\mathbf{A}-\mathbf{B})\). Now, let's expand this product using the distributive property: \((\mathbf{A}-\mathbf{B})\cdot(\mathbf{A}-\mathbf{B}) = \mathbf{A}\cdot\mathbf{A} - \mathbf{A}\cdot\mathbf{B} - \mathbf{B}\cdot\mathbf{A} + \mathbf{B}\cdot\mathbf{B}\)

Step 3: Perform the dot product and simplify

Next, we perform the dot product and rearrange the terms as follows: \(\mathbf{A}\cdot\mathbf{A} - \mathbf{A}\cdot\mathbf{B} - \mathbf{B}\cdot\mathbf{A} + \mathbf{B}\cdot\mathbf{B} = a^2 - (\mathbf{A}\cdot\mathbf{B} + \mathbf{B}\cdot\mathbf{A}) + b^2\) Since the dot product is commutative, i.e., \(\mathbf{A}\cdot\mathbf{B} = \mathbf{B}\cdot\mathbf{A}\), we can simplify the expression: \(a^2 - 2(\mathbf{A}\cdot\mathbf{B}) + b^2\)

Step 4: Obtain the Cosine Law

Now, recall the definition of the dot product: \(\mathbf{A}\cdot\mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos{\theta}\), where \(\theta\) is the angle between the vectors \(\mathbf{A}\) and \(\mathbf{B}\). We can substitute this definition into our expression: \(a^2 - 2(ab\cos{\theta}) + b^2\) Finally, we obtain the cosine law by equating the expression to the square of the length of the third side, c: \(c^2 = a^2 - 2(ab\cos{\theta}) + b^2\) And that's the cosine law in plane trigonometry.

Key Concepts

Plane Trigonometry

Plane trigonometry deals with the relationships between the sides and angles of triangles in a flat, two-dimensional plane. One of the foundational elements in this study is understanding the properties and relations of triangles through different trigonometric laws, such as the cosine law.

The cosine law, \[ c^2 = a^2 + b^2 - 2ab\cos{\theta} \] where \( c \) is the side opposite the angle \( \theta \), and \( a \) and \( b \) are the other two sides, is central in solving problems where traditional right-angle trigonometry (like the Pythagorean theorem) does not apply. This law not only assists in finding unknown side lengths or angles in a triangle, but it also paves the way for deeper exploration of the triangle's properties.

Understanding how to derive and apply the cosine law is crucial in plane trigonometry as it enhances problem-solving skills and broadens comprehension of geometric properties.

Dot Product

The dot product, also known as the scalar product, is a key operation in vector mathematics, pivotal in both physics and engineering. The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is computed as:

\[ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos{\theta} \] where \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \) respectively, and \( \theta \) is the angle between them.

This formula captures the idea that the dot product measures how much one vector extends in the direction of another. It is not only utilized in deriving the cosine law but finds applications in determining angles, projecting vectors, and even in defining orthogonal vectors. A key property of the dot product is its commutative nature, meaning \( \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} \), which simplifies many vector calculations. Understanding this allows for more complex geometric interpretations, including determining the angle between vectors and the orthogonality.

Vector Expansion

Vector expansion involves breaking down vector expressions into simpler terms, which simplifies calculations and aids in deeper understanding of vector interactions within a mathematical framework. The expansion of expressions like \[ (\mathbf{A} - \mathbf{B}) \cdot (\mathbf{A} - \mathbf{B}) \] requires applying distributive properties similar to algebraic multiplication.

This specific expansion uses the dot product: \[ \mathbf{A} \cdot \mathbf{A} - \mathbf{A} \cdot \mathbf{B} - \mathbf{B} \cdot \mathbf{A} + \mathbf{B} \cdot \mathbf{B} \] Simplifying these terms involves recognizing that the commutative property of the dot product combines like terms, resulting in \[ a^2 - 2(\mathbf{A} \cdot \mathbf{B}) + b^2 \], which is then related to the cosine law.

By expanding vectors, we transform complex problems into manageable pieces, enhancing both our calculations and interpretations of geometric relationships. This approach's power is evident when deriving laws like the cosine law, where clarity in understanding individual vector contributions and relationships is crucial.