Problem 30 - Precalculus with Limits

Question

Finding the Magnitude of a Vector In Exercises \(25-30\) , use the dot product to find the magnitude of u. $$\mathbf{u}=-21 \mathbf{i}$$

Step-by-Step Solution

Answer

The magnitude of the vector \(\mathbf{u}=-21 \mathbf{i}\) is 21.

Step 1: Understanding the vector magnitude

The magnitude of a vector, denoted as \(|\mathbf{u}|\) or \(||\mathbf{u}||\), is a positive scalar quantity representing the vector's length from the origin of the vector space. It is calculated by taking the dot product of the vector with itself, and then taking the square root of that quantity.

Step 2: Analyzing and preparing the provided data

We have been provided with the vector \(\mathbf{u}=-21 \mathbf{i}\). Since in this case, the vector is entirely in the \(\mathbf{i}\) direction (x-axis), the components in the \(\mathbf{j}\) and \(\mathbf{k}\) directions are zero. Thus, we can express the vector as \(\mathbf{u} = -21\mathbf{i} + 0\mathbf{j} + 0\mathbf{k}\).

Step 3: Calculating the dot product

The dot product of the vector \(\mathbf{u}\) with itself is calculated as \(\mathbf{u} \cdot \mathbf{u} = (-21)^2 + 0^2 + 0^2 = 441\).

Step 4: Finding the magnitude of the vector

Now, to find the magnitude of the vector \(\mathbf{u}\), we take the square root of this dot product. So, \(|\mathbf{u}| = \sqrt{\mathbf{u} \cdot \mathbf{u}} = \sqrt{441} = 21\). Do note that vector magnitudes are always positive or zero, since they represent distances; in this case, despite the vector \(\mathbf{u}\) pointing in the negative \(\mathbf{i}\) direction, its magnitude is 21, not -21.

Key Concepts

Dot Product

To comprehend how to calculate the magnitude of a vector, we must first understand the concept of the dot product. The dot product, also referred to as the scalar product, is a mathematical operation that takes two vectors and returns a single number, a scalar.
A dot product is performed by multiplying corresponding elements of the vectors and then summing these products. Mathematically, given two vectors \(\textbf{a} = a_1\textbf{i} + a_2\textbf{j} + a_3\textbf{k}\) and \(\textbf{b} = b_1\textbf{i} + b_2\textbf{j} + b_3\textbf{k}\), their dot product would be \(\textbf{a} \. \textbf{b} = a_1b_1 + a_2b_2 + a_3b_3\).
This operation is significant when dealing with vector magnitudes because, by taking the dot product of a vector with itself, we can easily find its length or magnitude. The magnitude of vector \(\textbf{a}\) can be determined as \(|\textbf{a}| = \sqrt{\textbf{a} \. \textbf{a}}\), which is the square root of the sum of the squares of its components.

Vector Components

In the study of vectors, it's essential to break them down into their components. Each vector can be represented as a combination of its parts along the fundamental axes of the space it's defined in – typically the x, y, and z axes represented by unit vectors \(\textbf{i}, \textbf{j}, \textbf{k}\), respectively.

Understanding Components

Vector components are projections of the vector onto these axis directions. For instance, a three-dimensional vector \(\textbf{v} = v_1\textbf{i} + v_2\textbf{j} + v_3\textbf{k}\) consists of the components \(v_1, v_2,\) and \(v_3\). Each component reflects how much of the vector is pointing in the direction of the respective axis. In a practical example, the vector provided in the exercise, \(\textbf{u} = -21\textbf{i}\), has a component \(-21\) along the x-axis and zero along the y and z axes, simplifying the process to find its magnitude.

Scalar Quantity

In contrast to vectors, a scalar quantity is a physical measurement that only has magnitude and no associated direction. Scalars are used to represent concepts that do not have an inherent direction, such as temperature, time, speed, or as seen in our exercise, the magnitude of a vector.

Scalar in Context of Magnitude

When calculating the magnitude of a vector, the result is always a scalar. This is important in the sense that it tells you how much of something there is without specifying where it is going. In the context of our exercise, when we find the magnitude of \(\textbf{u}\), the result of \(21\) is a scalar quantity. It conveys the 'size' of the vector, but not the fact that \(\textbf{u}\) is pointing in a negative direction along the x-axis. This distinction between scalar and vector quantities is fundamental in physics and engineering, allowing a clearer understanding of different physical concepts.