Problem 19 - Trigonometry

Question

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=-4 \mathrm{i}-3 \mathrm{j}, \mathbf{V}=-\mathrm{i}-2 \mathrm{j}\)

Step-by-Step Solution

Answer

The dot product \(\mathbf{U} \cdot \mathbf{V}\) is 10.

Step 1: Understand the Dot Product

The dot product of two vectors \(\mathbf{U} = a_1 \mathbf{i} + b_1 \mathbf{j}\) and \(\mathbf{V} = a_2 \mathbf{i} + b_2 \mathbf{j}\) is given by the formula \(\mathbf{U} \cdot \mathbf{V} = a_1a_2 + b_1b_2\). This involves multiplying corresponding components of the vectors and adding the results.

Step 2: Identify Components

Identify the components of the vectors \(\mathbf{U}\) and \(\mathbf{V}\). Here, \(\mathbf{U} = -4 \mathbf{i} - 3 \mathbf{j}\) has components \(a_1 = -4\) and \(b_1 = -3\). \(\mathbf{V} = -\mathbf{i} - 2 \mathbf{j}\) has components \(a_2 = -1\) and \(b_2 = -2\).

Step 3: Compute the Products

Multiply the corresponding components together: \(a_1a_2 = (-4)(-1) = 4\) and \(b_1b_2 = (-3)(-2) = 6\). These are the partial results we obtained by multiplying the corresponding components of the vectors.

Step 4: Add the Results

Add the products obtained in Step 3 to find the dot product. So, \(\mathbf{U} \cdot \mathbf{V} = 4 + 6 = 10\). This final value represents the dot product of the two vectors.

Key Concepts

Understanding Vectors

Vectors are fundamental elements in mathematics, especially in physics and engineering. They represent quantities that have both magnitude (how much) and direction (which way). An example of such a quantity is velocity, where you need to know not just how fast something is going, but also in what direction. A vector is often visualized as an arrow, where the length of the arrow indicates the magnitude, and the arrow's direction shows the way it points.
To describe a vector mathematically, we often use unit vectors. These are standard units, typically denoted as \(\mathbf{i}\) and \(\mathbf{j}\) in two-dimensional space, representing the directions along the x-axis and y-axis, respectively. For example, a vector \( \mathbf{U} = a \mathbf{i} + b \mathbf{j} \) might say, "go \(a\) units in the x-direction and \(b\) units in the y-direction."
This approach allows for an easy explanation and manipulation of vectors in calculations, especially when finding products like the dot product.

Vector Components Defined

Vector components are essentially the building blocks of any vector. Each vector can be broken down into two main parts when we're talking about two dimensions: the horizontal component and the vertical component. These components align with our basic unit vectors, \(\mathbf{i}\) for horizontal, and \(\mathbf{j}\) for vertical. So, a vector in the form of \(\mathbf{U} = a \mathbf{i} + b \mathbf{j}\) can be thought of as combining its effects in two separate dimensions.
Understanding the components is crucial for performing operations such as addition, subtraction, and particularly the dot product. When we deal with quantities like velocity or force, knowing the components allows us to analyze the vector in terms of its independent effect in each direction. For example, if you want to know purely how fast something is moving horizontally, you look at the \(a\) component. The vector \(\mathbf{U} = -4 \mathbf{i} - 3 \mathbf{j}\) tells us it moves 4 units left and 3 units down.

How to Perform the Dot Product Operation

The dot product is a special mathematical operation applied to two vectors. Unlike other operations like vector addition, which gives another vector, the dot product results in a scalar. This means you end up with just a number, no direction needed. This can be especially useful in physics when figuring out work done or projections.
The dot product formula takes the form \( \mathbf{U} \cdot \mathbf{V} = a_1a_2 + b_1b_2 \). You multiply the respective components of each vector and then add these results together. So if you had vectors \( \mathbf{U} = -4 \mathbf{i} - 3 \mathbf{j} \) and \( \mathbf{V} = -\mathbf{i} - 2 \mathbf{j} \), you would multiply the \(\mathbf{i}\) components \((-4)(-1) = 4\), and \(\mathbf{j}\) components \((-3)(-2) = 6\).
Adding them gives you \(10\), the dot product of \( \mathbf{U} \cdot \mathbf{V} = 10 \). This result tells you how much of one vector goes in the same direction as another, valuable in fields like physics and engineering for determining work or energy transfer.