Problem 47 - Precalculus with Limits

Question

Finding a Unit Vector In Exercises \(39-48,\) find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1 . . $$\mathbf{w}=\mathbf{i}-2 \mathbf{j}$$

Step-by-Step Solution

Answer

The unit vector in the direction of the vector \(\mathbf{w}=\mathbf{i}-2 \mathbf{j}\) is \(\frac{1}{\sqrt{5}}\mathbf{i} - \frac{2}{\sqrt{5}}\mathbf{j}\)

Step 1: Identify the given vector

The given vector is \(\mathbf{w}=\mathbf{i}-2 \mathbf{j}\)

Step 2: Calculate the magnitude of the given vector

The magnitude of a vector \(\mathbf{w} = a\mathbf{i} + b\mathbf{j}\) is calculated using the formula \(\sqrt{a^2 + b^2}\). Hence, the magnitude of the given vector \(\mathbf{w}=\mathbf{i}-2 \mathbf{j}\) = \(\sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4}\) = \(\sqrt{5}\)

Step 3: Find the unit vector

The unit vector in the direction of a given vector \(\mathbf{w}\) is obtained by dividing \(\mathbf{w}\) by its magnitude. Hence, the unit vector of \(\mathbf{w} = \frac{\mathbf{i} - 2\mathbf{j}}{\sqrt{5}}\) = \(\frac{1}{\sqrt{5}}\mathbf{i} - \frac{2}{\sqrt{5}}\mathbf{j}\)

Step 4: Verify the Result

Verify that the result has a magnitude of 1, by calculating the magnitude of the unit vector. The magnitude should equal to \(sqrt{(\frac{1}{\sqrt{5}})^2 + (-\frac{2}{\sqrt{5}})^2}\) = \(sqrt{1/5 + 4/5} = 1\)

Key Concepts

Vector Magnitude

Understanding vector magnitude is crucial in the field of vectors in mathematics. It represents the length or size of a vector, irrespective of its direction. The magnitude of a vector \textbf{v} can be symbolically written as \(\lVert \textbf{v} \rVert\).For a 2-dimensional vector \(\textbf{v} = a\textbf{i} + b\textbf{j}\), the magnitude is found using the Pythagorean theorem, resulting in the formula \(\sqrt{a^2 + b^2}\). This comes from the vector's interpretation as a right-angled triangle with horizontal and vertical components 'a' and 'b', respectively.In the context of the given exercise, the magnitude of vector \(\mathbf{w}=\(\mathbf{i}-2 \mathbf{j}\) \) is calculated as \(\sqrt{1^2 + (-2)^2} = \sqrt{5}\), which provides the necessary scalar value to proceed to vector normalization and find the unit vector.

Vector Normalization

Vector normalization is the process of converting a vector to a unit vector, which is a vector with a magnitude of 1 while preserving its direction. This achieves the standardization of vectors, which is particularly useful in various mathematical and physics applications where only the direction of the vector is of interest.To normalize a vector, you divide it by its magnitude. The formula is as follows: \( \textbf{u} = \frac{\textbf{v}}{\lVert \textbf{v} \rVert} \) where \( \textbf{u} \) is the unit vector and \( \textbf{v} \) is the original vector.For the example of vector \(\mathbf{w}\), the normalized vector (or unit vector) is \( \frac{1}{\sqrt{5}}\mathbf{i} - \frac{2}{\sqrt{5}}\mathbf{j} \) as shown in the exercise. Ensuring that the magnitude of this unit vector is 1 validates the normalization process.

Vectors in Mathematics

Vectors play an integral role in mathematics and its diverse applications, characterizing quantities with both magnitude and direction. This differentiates vectors from scalars, which only have magnitude. The study of vectors encompasses operations such as addition, subtraction, scalar multiplication, and dot product, to name a few.Vectors are denoted symbolically with arrows or boldface characters, represented in various dimensions using a coordinate system. The basic units along the coordinate axes are marked as \(\mathbf{i}, \mathbf{j}\), and sometimes \(\mathbf{k}\) for the third dimension.Vectors are utilized to describe physical phenomena such as velocity, force, and displacement. Their properties and operations facilitate solving complex problems in physics, engineering, and computer graphics, making their comprehension fundamental in STEM fields.

Vector Operations

Vector operations are a collection of mathematical tools that allow us to manipulate and combine vectors to explore their relations and produce new vectors. Key operations include vector addition and subtraction, which follow the 'head-to-tail' rule, allowing vectors to be combined based on their directional components.

Scalar Multiplication

Scalar multiplication involves scaling a vector by a real number, affecting its magnitude but not its direction. This is illustrated in the process of normalization where a vector is divided by its magnitude—a form of scalar division.

Dot Product

Another fundamental operation is the dot product, which yields a scalar value and is significant in determining the angle between two vectors.These operations, along with others like the cross product and vector projection, are the building blocks for more advanced topics in physics and engineering.