Problem 72 - Precalculus

Question

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

Step-by-Step Solution

Answer

The property \(k(\mathbf{u} \cdot \mathbf{v}) = (k \mathbf{u}) \cdot \mathbf{v} = \mathbf{u} \cdot (k \mathbf{v})\) is true.

Step 1: Calculate Scalar Multiplication of Dot Product

Given two vectors \(\mathbf{u} = \langle a, b \rangle\) and \(\mathbf{v} = \langle c, d \rangle\), the dot product is defined as \(\mathbf{u} \cdot \mathbf{v} = ac + bd\). When we perform scalar multiplication on the dot product by \(k\), the expression becomes \(k(\mathbf{u} \cdot \mathbf{v}) = k(ac + bd)\). Simplifying, this gives \(k(ac) + k(bd)\).

Step 2: Compute Dot Product of Scalar Multiplication on u

Next, consider the product \((k\mathbf{u}) \cdot \mathbf{v}\), where \(k\mathbf{u} = \langle ka, kb \rangle\). The dot product becomes \((k\mathbf{u}) \cdot \mathbf{v} = (ka)c + (kb)d = kac + kbd\). This matches the result from Step 1.

Step 3: Dot Product of Scalar Multiplication on v

Now compute \(\mathbf{u} \cdot (k\mathbf{v})\), where \(k\mathbf{v} = \langle kc, kd \rangle\). Then \(\mathbf{u} \cdot (k\mathbf{v}) = a(kc) + b(kd) = kac + kbd\). This outcome is consistent with the findings in Steps 1 and 2.

Step 4: Conclusion on Property Verification

Since all three calculations in Steps 1, 2, and 3 give us the same result, \(k(ac + bd)\), it is verified that \(k(\mathbf{u} \cdot \mathbf{v}) = (k \mathbf{u}) \cdot \mathbf{v} = \mathbf{u} \cdot (k \mathbf{v})\). Hence, the given property is true.

Key Concepts

Scalar Multiplication

Scalar multiplication in the context of vectors is a straightforward yet very important concept in linear algebra. It involves multiplying a vector by a scalar, which is simply a number. When you multiply a vector by a scalar, you scale the vector either up or down, depending on whether the scalar is greater or less than one, respectively.

Given a vector \(\mathbf{u} = \langle a, b \rangle \), multiplying by a scalar \( k \) results in \(k\mathbf{u} = \langle ka, kb \rangle\). This new vector points in the same direction as \( \mathbf{u} \) if \( k \) is positive, and the opposite direction if \( k \) is negative. The operation affects the length or magnitude of the vector but not its direction unless the scalar is negative.

Scalar multiplication is crucial when verifying properties of vectors, such as when checking how scalars interact with vector dot products.

Vector Properties

Understanding the diverse properties of vectors is foundational to mastering their applications. Vectors have distinct properties that govern operations such as addition and multiplication. The vector dot product, a critical operation, provides insights regarding angle relations and projection.

• Commutative Property: \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \). The dot product is unchanged when you swap the vectors.
• Distributive over Addition: \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \).
• Scalar Multiplication Property: The exercise in question focuses on verifying if \( k(\mathbf{u} \cdot \mathbf{v}) = (k \mathbf{u}) \cdot \mathbf{v} = \mathbf{u} \cdot (k \mathbf{v}) \). This demonstrates how a scalar impacts the interaction between two vectors when computing their dot product.

Getting a grip on these properties enables a more intuitive understanding of the behavior and nature of vectors in multidimensional spaces.

Dot Product Verification

Verifying the dot product properties involves demonstrating that certain algebraic manipulations yield consistent results across various approaches. The dot product of two vectors \(\mathbf{u} = \langle a, b \rangle\) and \(\mathbf{v} = \langle c, d \rangle\) is calculated as \(ac + bd\).

This exercise checked the property involving scalar multiplication of the dot product in three ways:

• First Approach: Multiply the entire dot product by the scalar: \( k(ac + bd) = kac + kbd \).
• Second Approach: Apply the scalar to the first vector before computing the dot product: \( (k \mathbf{u}) \cdot \mathbf{v} = \langle ka, kb \rangle \cdot \langle c, d \rangle = kac + kbd \).
• Third Approach: Apply the scalar to the second vector first: \( \mathbf{u} \cdot (k \mathbf{v}) = \langle a, b \rangle \cdot \langle kc, kd \rangle = a(kc) + b(kd) = kac + kbd \).

Each method converges to the same expression \(k(ac + bd) \), confirming the consistency and correctness of this vector property. This kind of verification underscores the robustness of mathematical operations on vectors, assuring that the properties remain reliable and valid.

Mastering such verification not only aids comprehension but also enhances the ability to apply vector operations accurately in various applications, from physics to computer graphics.