Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}\)

Step-by-Step Solution

Answer

The distributive property of the dot product is proven.

Step 1: Understand the Problem

We need to prove the distributive property of the dot product for two-dimensional vectors. This property states that the dot product of a vector with the sum of two other vectors is equal to the sum of the individual dot products with each vector.

Step 2: Define the Dot Product

The dot product of two vectors is calculated as follows: If \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), then \( \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 \).

Step 3: Express the Given Dot Product

Write the dot product \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \). First, express \( \mathbf{v} + \mathbf{w} \) as a new vector: \( \langle v_1 + w_1, v_2 + w_2 \rangle \). Then, compute \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \langle u_1, u_2 \rangle \cdot \langle v_1 + w_1, v_2 + w_2 \rangle \).

Step 4: Calculate the Dot Product

Apply the definition of the dot product to expand: \( u_1 \cdot (v_1 + w_1) + u_2 \cdot (v_2 + w_2) \). Simplify this expression to get: \( u_1 v_1 + u_1 w_1 + u_2 v_2 + u_2 w_2 \).

Step 5: Expand and Add Individual Dot Products

Now compute the individual dot products: \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 \) and \( \mathbf{u} \cdot \mathbf{w} = u_1 w_1 + u_2 w_2 \). Add these results: \( (u_1 v_1 + u_2 v_2) + (u_1 w_1 + u_2 w_2) \).

Step 6: Verify Equality of Expressions

We find that both expanded expressions from Step 4 and Step 5 are identical, and thus \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \) is proven.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector calculus. It takes two equal-length sequences of numbers (usually referred to as vectors) and returns a single number. The importance of the dot product in mathematics is deeply connected to its ability to represent projection. For two-dimensional vectors, think of the dot product as a way of measuring how much one vector goes in the direction of another vector. The formula for the dot product between two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 \). This resulting value is a scalar, hence the alternate name, scalar product.

• It's a measure of vectors' alignment - aligned vectors have a larger dot product.
• The result is positive if they point in a similar direction and negative if opposite.

Distributive Property

In mathematics, the distributive property is a fundamental principle that allows us to simplify and reorganize expressions. For vector calculus, particularly concerning the dot product, the distributive property states that the dot product distributes over vector addition. This property implies that for any three vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \), the expression \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \) holds true.

• Helps in simplifying complex vector calculations.
• Essential in proofs and derivations within vector calculus.

To visualize, consider \( \mathbf{u} \) being distributed across the sum of \( \mathbf{v} \) and \( \mathbf{w} \), like you would distribute a number across an addition inside parentheses. It maintains the balance of expressions and is crucial for vector transformations.

Two-Dimensional Vectors

Two-dimensional vectors are essential building blocks in vector calculus and are often used to model directions and magnitudes in a flat plane. A two-dimensional vector \( \mathbf{a} \) can be represented as \( \langle a_1, a_2 \rangle \), where \( a_1 \) and \( a_2 \) are its components along the x and y-axis, respectively. These vectors have a point of origin, usually at the coordinates (0,0) and stretch out to the coordinates \( (a_1, a_2) \). They can represent everything from simple movements to forces.

• Used in physics for depicting forces, velocities, or positions in a plane.
• Foundational in computer graphics for rendering 2D shapes.

Understanding the role and operations of two-dimensional vectors is fundamental to solving problems in various scientific and engineering fields.

Mathematical Proof

Mathematical proof provides a rigorous way of verifying the truth of statements and properties, ensuring that conclusions are logically sound based on given assumptions and axioms. In vector calculus, proofs often demonstrate that certain properties or formulas hold true under specific conditions. When you see a proof involving vectors, like demonstrating that the dot product distributes over addition, it guarantees that this is always valid for any set of two-dimensional vectors.

• Proofs in vector calculus often rely on algebraic manipulations and logical reasoning.
• They serve as a foundation for advanced mathematical concepts and real-world applications.

Proofs solidify your understanding of concepts, helping to build the bridge between abstract mathematical theories and practical applications. Getting comfortable with proofs equips students with critical thinking and problem-solving skills that are valuable across disciplines.