Problem 46 - 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 \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}\)

Step-by-Step Solution

Answer

The dot product is commutative: \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \).

Step 1: Understand the Dot Product

The dot product of two vectors \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\) is defined as \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2\). Similarly, the dot product \(\mathbf{v}\cdot\mathbf{u}\) is \(v_1u_1 + v_2u_2\). To prove the property, we need to show that these two expressions are equal.

Step 2: Perform Dot Product Calculations

Compute \(\mathbf{u} \cdot \mathbf{v}\) which equals \( u_1v_1 + u_2v_2 \). Now, compute \(\mathbf{v} \cdot \mathbf{u}\) which equals \( v_1u_1 + v_2u_2 \). Observe that \( u_1v_1 = v_1u_1 \) and \( u_2v_2 = v_2u_2 \) due to the commutative property of multiplication.

Step 3: Apply Commutative Property

Notice that multiplication is commutative, meaning that \( a \cdot b = b \cdot a \) for any real numbers \( a \) and \( b \). Applying this property to our dot products, we see that \( u_1v_1 + u_2v_2 = v_1u_1 + v_2u_2 \).

Step 4: Conclude the Proof

Having shown that \( u_1v_1 + u_2v_2 = v_1u_1 + v_2u_2 \), we conclude that \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \). Thus, the property is proven by demonstrating equality through the commutative nature of multiplication.

Key Concepts

Commutative Property

The commutative property is a fundamental concept in mathematics. It states that the order in which two numbers are multiplied does not affect the result. In formula terms, this means that for any two numbers, say \(a\) and \(b\), the equation \(a \cdot b = b \cdot a\) always holds true.

This property is crucial when working with vector dot products. A dot product involves multiplying corresponding components of two vectors and adding the results. By applying the commutative principle, we see that swapping the order of multiplication does not change the result.

In a simple example, if \( a = 3 \) and \( b = 4 \), then \( 3 \times 4 = 12 \) and \( 4 \times 3 = 12 \). The output remains consistent, affirming the commutative rule. For vector contexts, this translates directly into why \( \mathbf{u} \cdot \mathbf{v} \) equals \( \mathbf{v} \cdot \mathbf{u} \). Understanding the commutative property helps simplify calculations and ensures accuracy.

Two-Dimensional Vectors

Two-dimensional vectors are fundamental entities in mathematics and physics. They consist of two components, often represented as \(\mathbf{u} = \langle u_1, u_2 \rangle \). Each component corresponds to a dimension, one for the x-axis and the other for the y-axis in a coordinate system.

These vectors can represent various quantities, like force, velocity, or position, depending on the context. By breaking down a complex problem into vector components, you can analyze different dimensions separately, making problem-solving more manageable.

In our exercise, we have two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), each with two dimensions. The dot product of these vectors gives us valuable information about their interaction. Specifically, it measures how much the vectors align with each other.

• When the dot product is zero, the vectors are perpendicular.
• A positive dot product indicates vectors pointing in generally the same direction.
• A negative result means they point in opposite directions.

This simple application shows the utility of two-dimensional vectors in understanding physical and mathematical spaces.

Vector Notation

Vector notation is a concise and systematic way of representing vectors. It typically uses angle brackets to signify the components of the vector. For example, a vector can be written as \(\mathbf{a} = \langle a_1, a_2 \rangle\), where \(a_1\) and \(a_2\) are the horizontal and vertical components, respectively.

This notation is essential for clear communication in mathematics and science. Writing vectors in this form allows us to identify quickly and accurately the necessary components for calculations like addition, subtraction, and dot products.

In the context of the exercise, the vectors \(\mathbf{u}\), \(\mathbf{v}\), and potentially \(\mathbf{w}\) are expressed in vector notation. This representation simplifies our understanding of how vector operations are performed. It allows us to easily pick out components, \(u_1\) and \(u_2\), or \(v_1\) and \(v_2\), for use in calculations like dot products.

• Ensures clarity by separating vector components clearly.
• Facilitates the application of mathematical operations like the dot product.
• Improves the ability to generalize solutions across various mathematical or physical problems.

Using vector notation is a key skill in vector mathematics, aiding in making complex tasks simpler and more intuitive.