Problem 26 - College Trigonometry

Question

Approximate the component form of the vector \(\vec{v}\) using the information given about its magnitude and direction. Round your approximations to two decimal places. \(\|\vec{v}\|=392\); when drawn in standard position \(\vec{v}\) makes a \(117^{\circ}\) angle with the positive \(x\) -axis

Step-by-Step Solution

Answer

The vector \( \vec{v} \) in component form is \( \langle -178.13, 349.19 \rangle \).

Step 1: Understanding Components

The vector \( \vec{v} \) can be broken down into its components along the \( x \)-axis and \( y \)-axis. These components are related to the vector's magnitude and direction.

Step 2: Apply Trigonometry

Use trigonometric functions to find the components. The \( x \)-component of the vector is \( v_x = \|\vec{v}\| \cos(\theta) \), and the \( y \)-component is \( v_y = \|\vec{v}\| \sin(\theta) \), where \( \theta = 117^\circ \).

Step 3: Calculate the x-component

Calculate \( v_x = 392 \cos(117^\circ) \). Use a calculator to find \( \cos(117^\circ) \), which is approximately -0.45399. Thus, \( v_x \approx 392 \times (-0.45399) \approx -178.13 \).

Step 4: Calculate the y-component

Calculate \( v_y = 392 \sin(117^\circ) \). Use a calculator to find \( \sin(117^\circ) \), which is approximately 0.89101. Thus, \( v_y \approx 392 \times 0.89101 \approx 349.19 \).

Step 5: Write the Component Form

The approximate component form of \( \vec{v} \) is \( \vec{v} = \langle -178.13, 349.19 \rangle \).

Key Concepts

Magnitude and Direction

Understanding vectors begins with recognizing their magnitude and direction. The magnitude of a vector represents its size or length. For example, the vector in our exercise has a magnitude of 392. This magnitude indicates how long the vector is on a coordinate plane.
The direction of a vector, on the other hand, tells us what angle the vector makes with a reference line, usually the positive x-axis. The vector in the exercise forms an angle of 117 degrees with the positive x-axis.
Together, the magnitude and direction completely describe a vector, specifying not only how far it reaches but also where it is pointing. This combination is foundational for understanding how vectors operate in different contexts, such as physics or engineering.

Trigonometric Functions

Trigonometric functions are crucial for finding vector components. These functions relate the angles of a triangle to its side lengths and are key to breaking down vectors into their x and y components.
In our exercise, we use \[ \cos(\theta) \quad \text{and} \quad \sin(\theta) \] to find the components of the vector. Here, \( \theta \) is the angle of 117 degrees. Cosine helps us find the x-component, while sine helps us discover the y-component.

• Cosine (\( \cos \)): This function helps in calculating the proportion of the vector's magnitude that extends along the x-axis.
• Sine (\( \sin \)): This function aids in finding the proportion of the vector's magnitude that spreads along the y-axis.

Understanding these functions and how they work will enable you to see how vectors can be decomposed, analyzed, and utilized in various scenarios.

Component Form of Vectors

The component form of a vector provides a way to express the vector using two values: one for its x-direction and one for its y-direction. This form is especially useful for calculations involving vector addition, subtraction, or scaling.
In our exercise, we calculated the component form of the vector as \(\langle -178.13, 349.19 \rangle. \) This means the vector moves -178.13 units in the x-direction and 349.19 units in the y-direction.

• X-component (\( v_x \)): The calculated x-value of -178.13 indicates movement in the negative x-direction, which is expected given the angle of 117 degrees.
• Y-component (\( v_y \)): The y-value of 349.19 suggests significant movement upward, aligned with the sine calculation's positive outcome.

By understanding and calculating components, you can analyze how a vector behaves in a two-dimensional space, providing insights that are applicable in real-world situations like navigation and force decomposition.