Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(35-40:\) Prove the property if a and \(b\) are vectors and \(m\) is a real number. $$ (\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}-\mathbf{b})=\mathbf{a} \cdot \mathbf{a}-\mathbf{b} \cdot \mathbf{b} $$

Step-by-Step Solution

Answer

The property holds; LHS simplifies to RHS.

Step 1: Expand the LHS Expression

Start by expanding the left-hand side (LHS) of the equation: \[ (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}). \] Using the distributive property of the dot product, we can expand this to: \[ (\mathbf{a} \cdot \mathbf{a} - \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b}). \] Remember, for dot products, \( \mathbf{b} \cdot \mathbf{a} = \mathbf{a} \cdot \mathbf{b} \).

Step 2: Simplify the Expanded Expression

In the expanded expression \[ \mathbf{a} \cdot \mathbf{a} - \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b}, \] notice that the terms \( -\mathbf{a} \cdot \mathbf{b} \) and \( \mathbf{b} \cdot \mathbf{a} \) combine to zero because they are equal and opposite. Thus, we are left with \[ \mathbf{a} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b}. \]

Step 3: Conclude the Proof

The simplified expression, \( \mathbf{a} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b} \), matches the right-hand side (RHS) of the equation. Thus, we have shown that \[ (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b}. \] The property is proven.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It combines two vectors to produce a single number, known as a scalar. Given two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the dot product is calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).

This result is not a vector but a scalar quantity, hence the name scalar product. It is important to note that the dot product gives insight into the relationship between the two vectors:

• A dot product of zero implies that the vectors are perpendicular.
• A positive dot product suggests that the angle between the vectors is less than 90 degrees, indicating they point in a roughly similar direction.
• A negative dot product indicates an angle greater than 90 degrees, meaning the vectors point in somewhat opposite directions.

Understanding the dot product is essential as it frequently appears in various vector identities and properties.

Distributive Property

The distributive property is a crucial concept when dealing with operations in vector algebra. It allows the expansion of expressions involving the sum of vectors inside dot products. For vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \), the distributive property is written as:

• \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \).

This property ensures that we can rearrange and simplify vector equations. In the given problem, the expression \( (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \) is expanded using the distributive property. This allows us to handle each term individually:

• First, the dot product is distributed: \( \mathbf{a} \cdot \mathbf{a},\) \( -\mathbf{a} \cdot \mathbf{b},\) \( \mathbf{b} \cdot \mathbf{a}, \) and \( -\mathbf{b} \cdot \mathbf{b} \).
• Recognizing that \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \) is key, as those terms cancel each other out, simplifying the expression to \( \mathbf{a} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b} \).

Through the distributive property, we can see how combining vectors with other mathematical operations helps simplify and solve vector expressions.

Vector Identities

Vector identities are mathematical relationships that hold true for vectors under certain operations, like addition and the dot product. They are vital tools in simplifying and solving vector equations. Here are some useful vector identities that you might encounter:

• Self Dot Product: The dot product of a vector with itself is equal to its magnitude squared: \( \mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2 \).
• Orthogonal Vectors Dot Product: Two vectors are orthogonal (perpendicular) if and only if their dot product is zero.
• Dot Product Symmetry: The dot product is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).

These identities simplify vector calculations and are frequently used to prove properties like the one in the given exercise. Recognizing when and how to use these identities can make solving complex vector problems much more straightforward. This capability is crucial when interpreting physical situations or mathematical models involving vector spaces. As shown in the solution, understanding these vector identities allows us to tackle the problem methodically and achieve a clear, simplified result.