Problem 7 - APEX Calculus

Question

Find the dot product of the given vectors. \(\vec{u}=\langle 1,-1,2\rangle, \vec{v}=\langle 2,5,3\rangle\)

Step-by-Step Solution

Answer

The dot product is 3.

Step 1: Understand the Dot Product Formula

The dot product of two vectors \( \vec{u} = \langle a, b, c \rangle \) and \( \vec{v} = \langle x, y, z \rangle \) is given by the formula \( \vec{u} \cdot \vec{v} = ax + by + cz \). We will use this formula to find the dot product of \( \vec{u} = \langle 1, -1, 2 \rangle \) and \( \vec{v} = \langle 2, 5, 3 \rangle \).

Step 2: Substitute the Vector Components

Substitute the components of the vectors into the dot product formula: \( \vec{u} \cdot \vec{v} = 1 \cdot 2 + (-1) \cdot 5 + 2 \cdot 3 \).

Step 3: Calculate Each Product

Calculate the individual products: \( 1 \cdot 2 = 2 \), \( (-1) \cdot 5 = -5 \), and \( 2 \cdot 3 = 6 \).

Step 4: Sum the Results

Add the calculated products from the previous step: \( 2 + (-5) + 6 \).

Step 5: Compute the Final Dot Product

Calculate the sum: \( 2 - 5 + 6 = 3 \). Thus, the dot product of the vectors is 3.

Key Concepts

Understanding Vectors

Vectors are fundamental in mathematics and physics for expressing quantities with both magnitude and direction. Think of a vector as an arrow pointing from one location to another in space. It has a certain length (magnitude) and points in a particular direction. In mathematical terms, vectors are often represented as tuples of numbers indicating their position in space. For example, the vector \( \vec{u} = \langle 1, -1, 2 \rangle \) is positionally described in three-dimensional space by its components 1, -1, and 2.

• Magnitude: Measures the length of the vector.
• Direction: Defined by the components in the tuple.

Vectors are vital in various fields, such as engineering, physics, and computer graphics.

Breaking Down Vector Components

A vector component refers to each part of a vector that determines its position along a coordinate axis. In our case, the vectors \( \vec{u} = \langle 1, -1, 2 \rangle \) and \( \vec{v} = \langle 2, 5, 3 \rangle \) are composed of three components each.

• \( \vec{u} \) has components 1, -1, and 2, representing its measurement along the x, y, and z axes, respectively.
• Similarly, \( \vec{v} \) has components 2, 5, and 3 for the same axes.

Understanding these components is crucial because they are used directly in calculations like finding a dot product. They help in translating the vector's orientation and size into actionable mathematics.

Step-by-Step Calculation of the Dot Product

Calculating the dot product involves multiplying corresponding components of two vectors and summing those products. Let's break it down:1. Use the Real Need-to-Know Formula: The dot product \( \vec{u} \cdot \vec{v} \) is calculated as follows: \[ \vec{u} \cdot \vec{v} = a \times x + b \times y + c \times z \] where \( \vec{u} = \langle a, b, c \rangle \) and \( \vec{v} = \langle x, y, z \rangle \).2. Substitute the Vector Components: Insert the actual numbers from the vectors: \[ \vec{u} \cdot \vec{v} = 1 \times 2 + (-1) \times 5 + 2 \times 3 \]3. Calculate Each Product: Perform the multiplications: \( 1 \times 2 = 2 \), \( (-1) \times 5 = -5 \), and \( 2 \times 3 = 6 \).4. Sum the Results: Add these products together: \[ 2 + (-5) + 6 = 3 \]Thus, the dot product of vectors \( \vec{u} \) and \( \vec{v} \) is 3.

The Mathematics Behind It All

The dot product, also known as the scalar product, is a crucial concept in vector mathematics. It gives us a way to determine how much one vector extends in the direction of another. Interestingly, this results in a scalar quantity, not a vector.

• When two vectors are parallel, the dot product equals the product of their magnitudes.
• If they're perpendicular, the dot product is zero because they don’t "overlap" in any direction.

This mathematical property of vectors forms the backbone of many applications, including physics simulations, computer graphics, and even in optimization problems. By mastering the dot product, you gain deep insights into how vectors interact in n-dimensional spaces.