Problem 33 - Algebra and Trigonometry

Question

33-36 Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors and let \(a\) be a scalar. Prove the given property. $$\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$$

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 \( \mathbf{u} \) and \( \mathbf{v} \) denoted as \( \mathbf{u} \cdot \mathbf{v} \), is calculated as \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + ... + u_nv_n \) where the vectors are \( \mathbf{u} = \{u_1, u_2, ..., u_n\} \) and \( \mathbf{v} = \{v_1, v_2, ..., v_n\} \).

Step 2: Compute \(\mathbf{u} \cdot \mathbf{v}\)

Substitute the components of the vectors into the dot product formula: \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + ... + u_nv_n \).

Step 3: Compute \(\mathbf{v} \cdot \mathbf{u}\)

Similarly, compute the dot product but switch the roles of \( \mathbf{u} \) and \( \mathbf{v} \): \( \mathbf{v} \cdot \mathbf{u} = v_1u_1 + v_2u_2 + ... + v_nu_n \).

Step 4: Compare \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{v} \cdot \mathbf{u}\)

Notice that \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + ... + u_nv_n \) is exactly the same as \( \mathbf{v} \cdot \mathbf{u} = v_1u_1 + v_2u_2 + ... + v_nu_n \) since addition is commutative and \( u_iv_i = v_iu_i \) for each \( i \).

Step 5: Conclusion

Since the expressions for \( \mathbf{u} \cdot \mathbf{v} \) and \( \mathbf{v} \cdot \mathbf{u} \) are identical, we conclude that \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \). Thus, the dot product is commutative.

Key Concepts

Vectors

Vectors are essential in mathematics and physics because they help us describe quantities that have both magnitude and direction. A vector is typically represented as a directed line segment, but it can also be expressed as an ordered list of numbers, which are its components.

• Each vector has a dimension based on the number of its components, such as \( \mathbf{u} = \{u_1, u_2, ..., u_n\} \).
• The notation \( \mathbf{u} \) signifies that \( \mathbf{u} \) is a vector, and the braces represent the components of the vector.
• Vectors can be added together or subtracted, leading to a new vector whose components are computed element-wise.

Understanding vectors and their properties, like magnitude and direction, is crucial for working with the dot product, which is a scalar value results from the interaction of two numerical vectors.

Commutative Property

The commutative property is a fundamental principle in mathematics that states the order of addition or multiplication does not change the result. For vectors and scalars, this means:

• When working with scalar multiplication of numbers, both addition and multiplication are commutative. This allows for rearranging terms without changing the outcome, as in \( a + b = b + a \) and \( a \times b = b \times a \).
• In the context of vectors, especially concerning the dot product, the commutative property states that \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \).
• This property is derived from the fact that multiplication of real numbers is commutative, meaning \( u_iv_i = v_iu_i \) for any components of the vectors \( \mathbf{u} \) and \( \mathbf{v} \).

Therefore, the commutative property assures us that the scalar result of a dot product remains the same irrespective of the order of the vectors involved.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is a single real number. This operation alters the magnitude of the vector but not its direction:

• When a vector \( \mathbf{u} = \{u_1, u_2, ..., u_n\} \) is multiplied by a scalar \( a \), the resulting vector is \( a\mathbf{u} = \{a \cdot u_1, a \cdot u_2, ..., a \cdot u_n\} \).
• The effect of scalar multiplication is to stretch or compress the vector in its direction based on the scalar value, whether it is greater than, less than, or equal to 1.
• Despite changing the magnitude, scalar multiplication preserves the vector's direction unless the scalar is negative, in which case the vector's direction reverses.

Understanding scalar multiplication is vital when combined with operations like the dot product since it affects the overall magnitude of the interaction between vectors.