Problem 40 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The property \( (a \mathbf{u}) \cdot \mathbf{v} = a(\mathbf{u} \cdot \mathbf{v}) = \mathbf{u} \cdot (a \mathbf{v}) \) is true and proven by scalar factorization in dot products.

Step 1: Understand Scalar Multiplication and Dot Product

The property involves vectors and dot products in combination with scalar multiplication. Recall that the dot product of two vectors, \( \mathbf{u} \) and \( \mathbf{v} \), is defined as \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \) for a 3-D vector. Scalar multiplication involves multiplying each component of a vector by a scalar.

Step 2: Break Down the Expression (a \mathbf{u}) \cdot \mathbf{v}

Rewrite the expression using component-wise definition for the vectors \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \). The vector \( a \mathbf{u} = (a u_1, a u_2, a u_3) \) is obtained by multiplying each component of \( \mathbf{u} \) by \( a \). The dot product becomes \( (a u_1)v_1 + (a u_2)v_2 + (a u_3)v_3 \).

Step 3: Factor Out the Scalar from Dot Product

Factor out the scalar \( a \) from the components in \( (a u_1)v_1 + (a u_2)v_2 + (a u_3)v_3 \), giving \( a(u_1v_1 + u_2v_2 + u_3v_3) \). This represents the expression for \( a(\mathbf{u} \cdot \mathbf{v}) \).

Step 4: Analyze \mathbf{u} \cdot (a \mathbf{v})

Consider the expression \( \mathbf{u} \cdot (a \mathbf{v}) \). We have \( a \mathbf{v} = (a v_1, a v_2, a v_3) \), thus \( \mathbf{u} \cdot (a \mathbf{v}) = u_1(a v_1) + u_2(a v_2) + u_3(a v_3) \).

Step 5: Factor Out the Scalar Again

Similar to Step 3, factor out \( a \) from \( u_1(a v_1) + u_2(a v_2) + u_3(a v_3) \), resulting in \( a(u_1v_1 + u_2v_2 + u_3v_3) \), which matches \( a(\mathbf{u} \cdot \mathbf{v}) \).

Step 6: Write the Final Equality

Notice that both \( (a \mathbf{u}) \cdot \mathbf{v} \) and \( \mathbf{u} \cdot (a \mathbf{v}) \) simplify to the same expression, \( a(\mathbf{u} \cdot \mathbf{v}) \). Thus, the property \( (a \mathbf{u}) \cdot \mathbf{v} = a(\mathbf{u} \cdot \mathbf{v}) = \mathbf{u} \cdot (a \mathbf{v}) \) is proven.

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in vector algebra. It involves multiplying every component of a vector by a scalar value. This operation scales the vector without changing its direction (unless the scalar is negative or zero).
For instance, if you have a vector \( \mathbf{u} = (u_1, u_2, u_3) \) and a scalar \( a \), the scalar multiplication \( a \mathbf{u} \) would result in the vector \( (a u_1, a u_2, a u_3) \).

• Scalar multiplication can stretch or shrink a vector depending on the absolute value of the scalar.
• When scalar multiplication is applied, the entire vector is affected uniformly.

Understanding scalar multiplication is crucial for deeper insights into vector operations, such as dot products, which involve both scalar multiplication and summation.

Vector Algebra

Vector algebra combines different vector operations to study relationships and structures involving vectors. An essential operation in vector algebra is the dot product (or scalar product).
The dot product of two vectors \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \) is calculated as \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \).

• The dot product results in a scalar value, hence the alias "scalar product."
• It provides a measure of the extent to which two vectors point in the same direction.
• In geometry, the dot product is significant for finding angles between vectors.

Understanding vector algebra helps in solving complex problems by breaking down vector operations into simpler components through scalar multiplication and other vector operations.

Mathematical Proofs

Mathematical proofs are logical processes used to verify that particular statements or properties are true. In vector problems, proofs often involve algebraic manipulation and understanding of fundamental concepts like scalar multiplication and the dot product.
A typical proof in vector algebra may involve:

• Expressing vectors in terms of their components.
• Using properties of algebra to reorder and simplify expressions.
• Demonstrating equality between different expressions through step-by-step reasoning.

When proving vector properties, it's essential to follow a logical sequence. For instance, showing that \( (a\mathbf{u}) \cdot \mathbf{v} = a(\mathbf{u} \cdot \mathbf{v}) = \mathbf{u} \cdot (a\mathbf{v}) \) requires understanding component-wise operations, leveraging distribution laws, and recognizing the role of scalar multiplication.
Mathematical proofs not only confirm the truth but also enhance the understanding of how mathematical concepts interact in vector algebra. This structured approach equips students to tackle even more complex mathematical challenges.