Problem 26 - Precalculus Mathematics for Calculus

Question

Two vectors \(u\) and \(v\) are given. Find their dot product \(\mathbf{U}^{*} \mathbf{V}\). $$\mathbf{u}=\langle- 3,0,4\rangle, \quad \mathbf{v}=\left\langle 2,4, \frac{1}{2}\right\rangle$$

Step-by-Step Solution

Answer

The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \(-4\).

Step 1: Understanding Dot Product Formula

The dot product of two vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\) is calculated as: \[ \mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2 + u_3 \cdot v_3 \] This formula sums the products of the corresponding components of the vectors.

Step 2: Extracting Vector Components

Identify the components of vectors \(\mathbf{u}\) and \(\mathbf{v}\): - For \(\mathbf{u}\), the components are \(u_1 = -3\), \(u_2 = 0\), \(u_3 = 4\).- For \(\mathbf{v}\), the components are \(v_1 = 2\), \(v_2 = 4\), \(v_3 = \frac{1}{2}\).

Step 3: Calculating the Dot Product

Substitute the vector components into the dot product formula: \[\mathbf{u} \cdot \mathbf{v} = (-3)(2) + (0)(4) + (4)\left(\frac{1}{2}\right)\] Calculate each term: - \((-3)(2) = -6\) - \(0 \cdot 4 = 0\) - \(4 \cdot \frac{1}{2} = 2\)

Step 4: Summing the Products

Add the computed values to find the final result of the dot product: \[-6 + 0 + 2 = -4\]

Key Concepts

Vector Components

In order to understand vectors and their operations, it is essential to become familiar with the concept of vector components. Vectors are mathematical entities that have both magnitude and direction. They are often described in a coordinate system by their components.
For example, let's consider a 3-dimensional vector like \( \mathbf{u} = \langle -3, 0, 4 \rangle \). \( \mathbf{u} \) consists of three components: \( u_1 = -3, u_2 = 0, \) and \( u_3 = 4 \). Each component corresponds to a position along one of the axes in a 3D space.
The components essentially split the vector into distinct parts, showing how much it "moves" in each direction.

• A positive component indicates movement in the positive direction on an axis.
• A negative component shows movement in the negative direction.
• A zero component indicates no movement along that axis.

This breakdown into components is essential because it allows us to perform vector operations efficiently, such as addition, subtraction, and, as detailed next, the dot product.

Vector Operations

Vector operations delve into the mathematical manipulation of vectors, including addition, subtraction, and multiplication. One common operation is the dot product. The dot product is a way to multiply two vectors, resulting in a scalar (a single number), rather than another vector.
In the exercise, we examine two vectors: \( \mathbf{u} = \langle -3, 0, 4 \rangle \) and \( \mathbf{v} = \langle 2, 4, \frac{1}{2} \rangle \). Unlike vector addition or subtraction, which produce another vector, the dot product offers a way to gauge the extent to which two vectors align with each other. To perform this operation properly, you need to multiply corresponding components from each vector and then sum these products:
1. Multiply each component pair: \( u_1 \times v_1, u_2 \times v_2, \text{ and } u_3 \times v_3 \).
2. Add all these products together to get a single scalar value.
This operation is crucial in fields like physics and computer science for tasks involving projections and work calculations.

Mathematical Calculation

Understanding the mathematical calculation process of the dot product is crucial. This involves using the specific formula \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \) as shown in the exercise. Let's break it down further using the example vectors \( \mathbf{u} = \langle -3, 0, 4 \rangle \) and \( \mathbf{v} = \langle 2, 4, \frac{1}{2} \rangle \).
First, we multiply the first components: \( (-3) \times 2 = -6 \).
Next, multiply the second components: \( 0 \times 4 = 0 \).
Then, multiply the third components: \( 4 \times \frac{1}{2} = 2 \).

• The second product is zero because one of the factors is zero; hence, the contribution to the dot product from this pair is none.

Finally, add these results: \( -6 + 0 + 2 \).
Resulting in a dot product of \( -4 \).
This calculation process helps determine the directional relationship between vectors and is a pivotal concept in vector calculus and linear algebra.