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Magazine Chapter 8: Problem 10
From the given magnitude and direction in standard position, write the vector in component form. Magnitude: 7 , Direction: \(305^{\circ}\)
Short Answer
Expert verified
The vector in component form is approximately (4.02, -5.75).
Step by step solution
01
Understand the Given Data
We have a magnitude of 7 and a direction angle of \(305^{\circ}\). This angle is given in standard position, which means it is measured counterclockwise from the positive x-axis.
02
Recall the Vector Component Formulas
A vector's components can be found using the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\), where \(r\) is the magnitude and \(\theta\) is the direction angle.
03
Convert the Angle to a Usable Format
The angle given is \(305^{\circ}\). This places the vector in the fourth quadrant of the coordinate system. Ensure your calculator is set to degrees if you are using one.
04
Calculate the x-component
Using the formula \(x = r \cos \theta\), substitute \(r = 7\) and \(\theta = 305^{\circ}\). Calculate: \[x = 7 \cos 305^{\circ} = 7 \cos (-55^{\circ})\]The cosine of \(305^{\circ}\) is the same as the cosine of \(-55^{\circ}\) because cosine is the same for angles with that difference. Compute to get the numerical value.
05
Calculate the y-component
Using the formula \(y = r \sin \theta\), substitute \(r = 7\) and \(\theta = 305^{\circ}\). Calculate: \[y = 7 \sin 305^{\circ} = 7 \sin (-55^{\circ})\]The sine of \(305^{\circ}\) is the same as the sine of \(-55^{\circ}\) because it's in the fourth quadrant where sine is negative. Compute to get the numerical value.
06
Express the Vector in Component Form
Combine the calculated x and y components to write the vector in component form as \((x, y)\). For this problem, we obtain the vector \((4.02, -5.75)\) after calculating the trigonometric functions in Steps 4 and 5.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Functions
Trigonometric functions play a crucial role in breaking down vectors into their components. These functions, specifically sine (`sin`) and cosine (`cos`), help us relate angles to side lengths in right triangles. When dealing with vectors, this becomes handy as a vector can be seen as a hypotenuse of a triangle drawn in a coordinate plane. Thus:
- **Cosine** helps find the horizontal (x) component of the vector.
- **Sine** assists with the vertical (y) component.
Using the formulas:
- **Cosine** helps find the horizontal (x) component of the vector.
- **Sine** assists with the vertical (y) component.
Using the formulas:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
Coordinate System
The coordinate system is a framework for defining positions of points using a pair of numerical coordinates along axes. Generally, in a two-dimensional Cartesian plane, we have:
- The x-axis which runs horizontally.
- The y-axis which runs vertically.
Vectors in this system have both direction and magnitude, often angled from the positive x-axis known as the standard position. In this context, direction angles are measured counterclockwise:
- The x-axis which runs horizontally.
- The y-axis which runs vertically.
Vectors in this system have both direction and magnitude, often angled from the positive x-axis known as the standard position. In this context, direction angles are measured counterclockwise:
- Standard position assists in identifying which quadrant the vector falls into.
- It provides a consistent method to locate vectors regardless of origin location.
Magnitude
Magnitude represents the size or length of a vector. It tells you how much vector there is, but not the direction. Think of it as the strength being exerted:
- A magnitude of 7 units can be visualized as stretching 7 units along a direct line path.
- In vectors, it's always a positive value representing quantity and not negative features.
Direction Angle
A direction angle in vectors gives the orientation of the vector relative to a reference direction, commonly the positive x-axis. Here鈥檚 why it matters:
- It tells us which way the vector is pointing.
- The standard position measures this angle counterclockwise from the positive x-axis.
