Problem 12 - APEX Calculus

Question

Create your own vectors \(\vec{u}\) and \(\vec{v}\) in \(\mathbb{R}^{3}\) and scalar \(c\) and show that \(c(\vec{u} \cdot \vec{v})=\vec{u} \cdot(c \vec{v})\)

Step-by-Step Solution

Answer

The equality \(c(\vec{u} \cdot \vec{v}) = \vec{u} \cdot (c \vec{v})\) holds true.

Step 1: Define Vectors and Scalar

Let's choose two vectors in \(\mathbb{R}^3\). Let \(\vec{u} = \langle 1, 2, 3 \rangle\) and \(\vec{v} = \langle 4, 5, 6 \rangle\). Also, choose a scalar \(c = 2\).

Step 2: Calculate the Scalar Multiplication of the Dot Product

First, compute the dot product \(\vec{u} \cdot \vec{v}\). The dot product is calculated as: \(\vec{u} \cdot \vec{v} = 1 \times 4 + 2 \times 5 + 3 \times 6 = 32\).Next, multiply the result by the scalar \(c\): \(c(\vec{u} \cdot \vec{v}) = 2 \times 32 = 64\).

Step 3: Scalar Multiplication with the Second Vector

Multiply the vector \(\vec{v}\) by the scalar \(c\):\(c \vec{v} = 2 \times \langle 4, 5, 6 \rangle = \langle 8, 10, 12 \rangle\).

Step 4: Calculate the Dot Product with the Scaled Vector

Now, compute the dot product of \(\vec{u}\) with the scaled vector \(c \vec{v}\):\(\vec{u} \cdot (c \vec{v}) = 1 \times 8 + 2 \times 10 + 3 \times 12 = 64\).

Step 5: Compare the Results

Both calculations gave the result \(64\). Therefore, we have shown that \(c(\vec{u} \cdot \vec{v}) = \vec{u} \cdot (c \vec{v})\).

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a single number, known as a scalar. This operation scales the magnitude of the vector without changing its direction. Take, for example, the vector \( \vec{v} = \langle 4, 5, 6 \rangle \). If we multiply every component of \( \vec{v} \) by the scalar \( c = 2 \), each component is scaled, resulting in a new vector: \( c \vec{v} = \langle 8, 10, 12 \rangle \).

• Identity property: Multiplying a vector by 1 gives the same vector.
• Zero vector: Multiplying by 0 results in the zero vector.
• Distributive property: Scalar multiplication distributes over vector addition.

This fundamental operation enables us to manipulate vectors and is critical in various vector operations.

Vector Operations

Working with vectors involves a variety of operations that allow us to analyze their properties and interactions. The essential operations include addition, subtraction, and multiplication (like the dot product).
When you add two vectors, say, \( \vec{u} = \langle 1, 2, 3 \rangle \) and \( \vec{v} = \langle 4, 5, 6 \rangle \), you add their corresponding components:

• Addition: \( \vec{u} + \vec{v} = \langle 1+4, 2+5, 3+6 \rangle = \langle 5, 7, 9 \rangle \)

Another operation is vector subtraction, which works similarly but involves subtracting the components instead.
Multiplication, such as the dot product, combines vectors to yield a scalar. Understanding these operations helps in transitioning from abstract concepts to practical applications in physics and engineering.

Properties of Dot Product

The dot product is a key operation in vector algebra, often used to find the angle between vectors or to project one vector onto another. It results in a scalar, not a vector.
How the dot product works: If \( \vec{u} = \langle 1, 2, 3 \rangle \) and \( \vec{v} = \langle 4, 5, 6 \rangle \), the dot product is calculated as: \( \vec{u} \cdot \vec{v} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32 \).Key properties include:

• Commutative Property: \( \vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u} \)
• Distributive over Addition: \( \vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w} \)
• Scalar Multiplication: \( c(\vec{u} \cdot \vec{v}) = \vec{u} \cdot (c \vec{v}) \).

These properties simplify vector calculations and are vital to solving problems in linear algebra and calculus.