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Chapter 8: Problem 9

From the given magnitude and direction in standard position, write the vector in component form. Magnitude: 8 , Direction: \(220^{\circ}\)

Short Answer

Expert verified
The vector in component form is \( \langle -6.128, -5.1424 \rangle \).

Step by step solution

01

Understand the Problem

The problem involves converting a vector from its magnitude and direction form to its component form. The given magnitude is 8, and the direction is \(220^{\circ}\).
02

Recall the Formula for Component Form

The component form of a vector \( \vec{v} \) can be expressed using its magnitude \( |\vec{v}| \) and direction \( \theta \) as: \[ \vec{v} = |\vec{v}| \cdot \langle \cos(\theta), \sin(\theta) \rangle \] This means the vector \( \vec{v} \) can be written as \( \langle |\vec{v}| \cos(\theta), |\vec{v}| \sin(\theta) \rangle \).
03

Calculate the Component for the X-axis

To find the x-component of the vector, we use the cosine function: \( \text{x-component} = |\vec{v}| \cdot \cos(\theta) = 8 \cdot \cos(220^{\circ}) \). Calculate \( \cos(220^{\circ}) \) using a calculator: \( \cos(220^{\circ}) \approx -0.7660 \). Therefore, the x-component is: \( 8 \cdot (-0.7660) = -6.128 \).
04

Calculate the Component for the Y-axis

To find the y-component of the vector, we use the sine function: \( \text{y-component} = |\vec{v}| \cdot \sin(\theta) = 8 \cdot \sin(220^{\circ}) \). Calculate \( \sin(220^{\circ}) \) using a calculator: \( \sin(220^{\circ}) \approx -0.6428 \). Therefore, the y-component is: \( 8 \cdot (-0.6428) = -5.1424 \).
05

Write the Vector in Component Form

Combine the x-component and y-component to write the vector in component form: \( \vec{v} = \langle -6.128, -5.1424 \rangle \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Magnitude and Direction
Every vector in a plane can be defined by two key aspects: magnitude and direction.
Magnitude refers to the length or size of the vector鈥攖he greater the magnitude, the longer the vector. We denote the magnitude of a vector by the symbol \(|\vec{v}|\). In our example, the magnitude is given as 8.
Direction indicates where the vector is pointing relative to a standard position. This is usually expressed in degrees or radians, starting from the positive x-axis and moving counterclockwise.
In our exercise, the direction is given as \(220^{\circ}\). This direction is standard, meaning it's measured from the positive x-axis towards the vector.
  • Magnitude: The length or size of the vector.
  • Direction: The angle at which the vector points.
Using Trigonometric Functions to Find Components
To convert a vector's magnitude and direction into its component form, we employ trigonometric functions: cosine and sine.
These functions help us determine the vector's x-component and y-component by projecting it onto the x-axis and y-axis respectively.

Component Formula:
  • Cosine function: Determines the x-component using the formula \(|\vec{v}| \cdot \cos(\theta)\).
  • Sine function: Determines the y-component using the formula \(|\vec{v}| \cdot \sin(\theta)\).
In our example, the direction \(220^{\circ}\) implies that:
  • For the x-component: \(8 \cdot \cos(220^{\circ}) \approx -6.128\)
  • For the y-component: \(8 \cdot \sin(220^{\circ}) \approx -5.1424\)
Trigonometric functions, thus, allow us to break down the vector into its horizontal and vertical movements.
Exploring the Coordinate System
Vectors are plotted on a coordinate system which consists of an x-axis (horizontal) and a y-axis (vertical).
This system helps visually represent both the magnitude and the direction of vectors by placing them in a 2D plane.
When a vector is expressed in component form, \(\langle x, y \rangle\), it tells us:
  • The x-component (first number in the bracket) shows the horizontal distance from the origin.
  • The y-component (second number in the bracket) shows the vertical distance from the origin.
In our step-by-step solution, the vector becomes \(\langle -6.128, -5.1424 \rangle\), represented as a point on the coordinate system, indicating movement left and downward from the origin.
  • The negative signs suggest the vector is in the third quadrant, where both x and y are negative.
Understanding the coordinate system is vital, as it shows the entire journey of a vector's movement from the initial point to its destination.

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