Problem 69 - Precalculus

Question

Given \(u=\langle a, b\rangle\) and \(v=\langle c, d\rangle,\) show that the following properties are true: $$\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$$

Step-by-Step Solution

Answer

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

Step 1: Define the dot product

The dot product of two vectors \( u = \langle a, b \rangle \) and \( v = \langle c, d \rangle \) is defined by the formula \( u \cdot v = ac + bd \).

Step 2: Calculate \( u \cdot v \)

Using the definition of the dot product, calculate \( u \cdot v = ac + bd \).

Step 3: Calculate \( v \cdot u \)

Similarly, calculate the dot product of \( v \) with \( u \) using the definition: \( v \cdot u = ca + db \).

Step 4: Compare the expressions

Notice that \( ac + bd \) is the same as \( ca + db \) due to the commutative property of multiplication, i.e., \( ac = ca \) and \( bd = db \).

Step 5: Conclude the proof

Since both expressions \( ac + bd \) and \( ca + db \) are equal, we have \( u \cdot v = v \cdot u \). This shows that the dot product is commutative.

Key Concepts

Vectors in Mathematics

Vectors are fundamental components in the field of mathematics and physics, representing quantities that have both magnitude and direction. A vector can be visualized as an arrow; the length of the arrow represents the magnitude and the direction indicates where it is pointing. For example, the vector \( u = \langle a, b \rangle \) can be thought of as a direction from the origin to the point \((a, b)\) in a two-dimensional plane.
Vectors can be added together and multiplied by scalars, operations that are useful in many mathematical applications.
One of the key products of vectors is the dot product, also known as the scalar product, which results in a scalar quantity. Understanding vectors lays the groundwork for mastering the dot product and its properties.

Understanding the Commutative Property

The commutative property is a fundamental property of many mathematical operations. It states that changing the order of the operands does not change the result.
This property applies to both addition and multiplication. For example, in addition, \( a + b = b + a \), and in multiplication, \( ab = ba \).
In the context of vectors, the dot product of two vectors \( u = \langle a, b \rangle \) and \( v = \langle c, d \rangle \) is said to be commutative because \( u \cdot v = v \cdot u \). This means that the scalar result of the dot product remains the same regardless of the order in which the vectors are multiplied; essential for vector algebra and crucial for theoretical and practical applications.

Proof of the Dot Product's Commutative Property

To prove that the dot product of vectors is commutative, consider the vectors \( u = \langle a, b \rangle \) and \( v = \langle c, d \rangle \).
The dot product is given by the formula \( u \cdot v = ac + bd \).
Calculating \( v \cdot u \), we use the formula to find \( v \cdot u = ca + db \).
By breaking down each term, we observe that \( ac = ca \) and \( bd = db \) due to the commutative nature of multiplication. Thus, \( ac + bd \) equals \( ca + db \), confirming that \( u \cdot v = v \cdot u \). This proof illustrates that the dot product is invariant to the order in which the vectors are multiplied, highlighting the commutative property of the dot product in vector algebra.