Problem 79 - Algebra and Trigonometry

Question

Use the vectors $$\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad \text { and } \quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j}$$ to prove the given property. $$ \mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u} $$

Step-by-Step Solution

Answer

The dot product is commutative; namely, for any two vectors \(\mathbf{u}\) and \(\mathbf{v}\), \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\). This is proven by expanding the dot products in terms of their individual components.

Step 1: Understand the Vector Dot Product

The dot product (also known as the scalar product) of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by \( u \cdot v = |u||v| \cos(\theta) \) where |u| and |v| are the magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\), and \(\theta\) is the angle between the two vectors.

Step 2: Proof Step 1: Express \(\mathbf{u}\) and \(\mathbf{v}\) in Terms of Components

Given the vectors \(\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}\) and \(\mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}\) where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors along the x and y axis, respectively.

Step 3: Proof Step 2: Compute the Dot Product

The dot product \(\mathbf{u} \cdot \mathbf{v}\) can be computed as follows: \(\mathbf{u} \cdot \mathbf{v} = a_{1}*a_{2} + b_{1}*b_{2} \). Similarly, the dot product \(\mathbf{v} \cdot \mathbf{u}\) is: \(\mathbf{v} \cdot \mathbf{u} = a_{2}*a_{1} + b_{2}*b_{1} \).

Step 4: Proof Step 3: Prove the Commutative Property

From the above steps, it is clear that \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\), because multiplication is commutative, i.e. \(a*b = b*a\) for all \(a\) and \(b\). Thus, the property holds for the given vectors.

Key Concepts

Commutative Property of Dot Product

The commutative property of the dot product is a fundamental characteristic that states the order in which two vectors are dotted does not affect the result.

To understand this, consider two vectors \( \mathbf{u} \) and \( \mathbf{v} \), which are represented in component form. If we calculate \( \mathbf{u} \cdot \mathbf{v} \) using the components, we determine this to be \( a_1a_2 + b_1b_2 \). When we reverse the order, calculating \( \mathbf{v} \cdot \mathbf{u} \), the result is \( a_2a_1 + b_2b_1 \), which, given the commutative nature of multiplication \( (a \times b = b \times a) \), shows that these two expressions are equal.

Therefore, no matter which vector comes first in the operation, the dot product will remain the same, thus \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \). The simplicity of this property is crucial when working with vector dot products, as it allows for flexibility in computations and simplifies proofs in vector calculus and physics.

Components of a Vector

Every vector in a Cartesian coordinate system can be broken down into its components along the axes of the system. These components are projections of the vector on the axes, which are typically represented by unit vectors.

Take \( \mathbf{u} = a_1 \mathbf{i} + b_1 \mathbf{j} \) for example. Here, \( a_1 \) and \( b_1 \) represent the vector's magnitude in the x and y direction respectively, while \( \mathbf{i} \) and \( \mathbf{j} \) represent the unit vectors along the x and y axes. This breakdown into components is incredibly useful because it allows us to perform vector operations such as addition, subtraction, and dot product calculations using simple algebra.

Magnitude of a Vector

The magnitude of a vector, often denoted as \( |\mathbf{v}| \) or \( ||\mathbf{v}|| \), represents the vector's length or size. It is calculated by taking the square root of the sum of the squares of its components.

For instance, the magnitude of vector \( \mathbf{v} = a_2 \mathbf{i} + b_2 \mathbf{j} \) is \( |\mathbf{v}| = \sqrt{a_2^2 + b_2^2} \).

Understanding the magnitude is crucial when working with vectors because it can represent physical quantities such as force or velocity. Moreover, when calculating the dot product, knowing the magnitudes of the vectors involved allows the use of the geometric definition \( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta) \), connecting the algebraic and geometric interpretations of vector multiplication.

Angle Between Vectors

The angle between two vectors is an important concept in understanding their geometric relationship. For two vectors \( \mathbf{u} \) and \( \mathbf{v} \), the cosine of the angle \( \theta \) between them can be obtained using the dot product formula: \( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta) \).

To find \( \theta \) itself, one can rearrange the equation: \( \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \), and then take the inverse cosine. It's worth noting that this relationship is only valid when neither vector is the zero vector since the angle is undefined in such a case.

The angle provides insight into the direction of vectors relative to one another and is used in various physics applications, such as determining whether two forces are acting perpendicular to each other or finding the work done by a force moving an object in the direction of displacement.