Problem 40 - College Trigonometry

Question

For the given vector \(\vec{v}\), find the magnitude \(\|\vec{v}\|\) and an angle \(\theta\) with \(0 \leq \theta<360^{\circ}\) so that \(\vec{v}=\|\vec{v}\|\langle\cos (\theta), \sin (\theta)\rangle\) (See Definition 11.8.) Round approximations to two decimal places. $$ \vec{v}=\langle 0, \sqrt{7}\rangle $$

Step-by-Step Solution

Answer

The magnitude of \(\vec{v}\) is approximately 2.65, and the angle \(\theta\) is \(90^\circ\).

Step 1: Find the magnitude of the vector

The magnitude of the vector \( \vec{v} = \langle 0, \sqrt{7} \rangle \) is given by \( \|\vec{v}\| = \sqrt{0^2 + (\sqrt{7})^2} = \sqrt{7} \). The calculated magnitude is \( \sqrt{7} \approx 2.65 \) when rounded to two decimal places.

Step 2: Express the vector in angle form

The vector \( \vec{v} = \langle 0, \sqrt{7} \rangle \) lies on the y-axis since its x-component is zero. Since the vector points upwards along the y-axis, the angle \( \theta \) with respect to the positive x-axis is \( 90^\circ \), because the vector is entirely in the positive y-direction.

Step 3: Formulate the vector using magnitude and angle

Using the magnitude \( \|\vec{v}\| = \sqrt{7} \) and angle \( \theta = 90^\circ \), the vector can be re-expressed as \( \vec{v} = \sqrt{7} \langle \cos(90^\circ), \sin(90^\circ) \rangle = \sqrt{7} \langle 0, 1 \rangle = \langle 0, \sqrt{7} \rangle \), confirming the correct formulation.

Key Concepts

Trigonometric Form of Vectors

Vectors are mathematical objects characterized by both magnitude and direction. In trigonometric form, a vector is expressed using trigonometric functions like cosine and sine, which outline its angle relative to a reference axis. This allows vectors to be represented in terms of their angles, simplifying analysis and calculation.

• The trigonometric form of a vector \( \vec{v} \) is expressed as \( \vec{v} = \|\vec{v}\| \langle \cos(\theta), \sin(\theta) \rangle \), where \( \|\vec{v}\| \) is the magnitude of the vector, and \( \theta \) is the angle with the positive x-axis.
• The cosine component, \( \cos(\theta) \), determines the projection of the vector along the x-axis, while the sine component, \( \sin(\theta) \), determines the projection along the y-axis.

Understanding the trigonometric form is essential as it bridges geometric and algebraic vector properties, allowing you to easily switch between vector forms and analyze their behavior.

Vector Angle

The angle \( \theta \) in a vector is pivotal as it defines the vector's direction. When you know a vector's magnitude and angle, you can determine its components.

• The vector angle \( \theta \) is measured from the positive x-axis, moving counterclockwise around a circle.
• In the context of the Cartesian plane, if a vector is purely vertical, like \( \vec{v} = \langle 0, \sqrt{7} \rangle \), the angle \( \theta \) is either \( 90^\circ \) or \( 270^\circ \), depending on whether it points up or down respectively. In this exercise, the vector points upward, giving \( \theta = 90^\circ \).

Determination of the angle helps in converting vectors between different forms and is critical in applications like physics and engineering where direction is essential.

Unit Vector

A unit vector is a vector that has a magnitude of exactly 1 and indicates direction only. Converting vectors into unit vectors can help standardize calculations and simplify vector operations.

• To convert a vector to a unit vector, divide each component of the vector by its magnitude \( \|\vec{v}\| \).
• If we take a vector like \( \vec{v} = \langle 0, \sqrt{7} \rangle \), its magnitude is \( \sqrt{7} \). The corresponding unit vector would be \( \langle 0, \frac{\sqrt{7}}{\sqrt{7}} \rangle = \langle 0, 1 \rangle \).

Unit vectors are very useful in defining direction without worrying about the vector's actual length, which is a common requirement in vector-based mathematical and physical calculations, such as normalizing a direction.