Problem 54 - Thomas Calculus

Question

Unit vectors in the plane Show that a unit vector in the plane can be expressed as \(\mathbf{u}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j},\) obtained by rotating \(\mathbf{i}\) through an angle \(\theta\) in the counterclockwise direction. Explain why this form gives every unit vector in the plane.

Step-by-Step Solution

Answer

A unit vector in the plane can be written as \( (\cos \theta) \mathbf{i} + (\sin \theta) \mathbf{j} \) by varying \( \theta \) from 0 to \( 2\pi \), representing all directions.

Step 1: Understanding Unit Vectors

A unit vector is a vector with a magnitude of 1. In the plane, these vectors can point in any direction but always have the same magnitude equal to 1.

Step 2: Expressing the Unit Vector

A vector can be expressed in terms of its components along the x-axis and y-axis. In the plane, let \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \) represent a unit vector, where \( a \) and \( b \) are real numbers.

Step 3: Using Magnitude Condition

Since \( \mathbf{u} \) is a unit vector, its magnitude is 1, i.e., \( ||\mathbf{u}|| = \sqrt{a^2 + b^2} = 1 \). This implies \( a^2 + b^2 = 1 \).

Step 4: Trigonometric Representation

Recognize that \( a \) and \( b \) can be represented as sine and cosine of an angle \( \theta \). Set \( a = \cos \theta \) and \( b = \sin \theta \), which satisfies \( a^2 + b^2 = 1 \) since \( \cos^2 \theta + \sin^2 \theta = 1 \).

Step 5: Form of the Unit Vector

Substitute \( a \) and \( b \) back into the unit vector equation: \( \mathbf{u} = (\cos \theta) \mathbf{i} + (\sin \theta) \mathbf{j} \). This shows any unit vector in the plane can be represented in this form by choosing an appropriate \( \theta \).

Step 6: Explanation of Rotation

By varying \( \theta \) from 0 to \( 2\pi \), every possible direction for the unit vector is covered, as this represents a full rotation around the circle. The starting vector \( \mathbf{i} \) (or x-axis direction) is rotated counterclockwise by angle \( \theta \), covering all possible unit vectors.

Key Concepts

Vector Magnitude

In the world of vectors, understanding the concept of vector magnitude is essential. The **magnitude** of a vector represents its size or length. For a vector in a 2D plane, denoted by \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \), the magnitude is calculated using the formula: \[||\mathbf{u}|| = \sqrt{a^2 + b^2}\]This formula comes from the Pythagorean theorem in geometry, which helps us determine the hypotenuse (or length) when the other two side lengths are known. When we talk about unit vectors, their magnitude is always equal to 1, making them a standardized way to express direction.Key points about **vector magnitude**:

• A unit vector has a magnitude of 1, no matter its direction.
• In mathematical notation, the magnitude of a vector \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \) is expressed as \( \sqrt{a^2 + b^2} = 1 \).
• This condition is crucial since it ensures that the vector represents purely direction without stretching or shrinking.

Unit vectors on the plane maintain this condition by ensuring that the square of the cosine and sine components always add up to one, symbolizing a point lying on the unit circle.

Trigonometric Representation

**Trigonometric representation** plays a vital role in expressing vectors, particularly unit vectors. A unit vector in a 2D plane can be represented using trigonometric functions, leveraging the natural properties of circular trigonometry. By setting the vector components as trigonometric functions of an angle \(\theta\), we derive the equation: \[\mathbf{u} = (\cos \theta) \mathbf{i} + (\sin \theta) \mathbf{j}\]This representation involves two primary trigonometric functions:

• \(\cos \theta\), which provides the horizontal component.
• \(\sin \theta\), which gives the vertical component.

These components form the legs of a right triangle where the hypotenuse is the radius of a unit circle, ensuring the magnitude is always 1.Why trigonometric representation works:

• Trigonometric identities, such as \( \cos^2 \theta + \sin^2 \theta = 1 \), guarantee that the sum of the squares of these components equals 1.
• It offers a continuous representation of all possible directions by varying \(\theta\), which ranges from 0 to \(2\pi\).

This approach effortlessly illustrates how a unit vector rotates counterclockwise around the origin, seamlessly adopting new directions while staying on the unit circle comprised of all directions a unit vector can possibly point towards.

Cartesian Coordinates

Vectors in two-dimensional space are often represented in **Cartesian coordinates**. This system divides the plane into horizontal (x-axis) and vertical (y-axis) components using pairs of numerical values. When describing a vector like \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \), \(a\) represents the x-component and \(b\) the y-component.**Connecting with Cartesian Coordinates:**

• Units along the x and y axes are often expressed using base vectors \( \mathbf{i} \) and \( \mathbf{j} \). These are the unit vectors pointing in the direction of the x-axis and y-axis, respectively.
• Unit vectors arranged in this way—like \( \mathbf{u} = (\cos \theta) \mathbf{i} + (\sin \theta) \mathbf{j} \)—still fall under Cartesian coordinates since you mention explicit values for each axis.

In essence, Cartesian coordinates simplify understanding of vectors by breaking them down into manageable horizontal and vertical components. With this coordinate system, any point or vector in the plane can be reached by appropriately scaling the unit vectors along these axes, essentially describing the object's position as a unique combination of its basic components. This breakout ensures clarity, as each vector holds numerically distinguishable, concrete positions in the 2D space, making them easier to analyze and understand.