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Magazine Chapter 8: Problem 56
Determine the direction angle \(\theta\) of the vector, to the nearest degree. $$\mathbf{w}=5 \mathbf{i}-\mathbf{j}$$
Short Answer
Expert verified
The direction angle is 349掳.
Step by step solution
01
Understand the vector components
The given vector \(\textbf{w}=5\textbf{i}-\textbf{j}\) has components \(w_{x}=5\) and \(w_{y}=-1\).
02
Use the tangent function
For a vector \(\textbf{w} = w_{x}\textbf{i} + w_{y}\textbf{j}\), the direction angle \(\theta\) with the positive x-axis is given by the formula \(\tan(\theta) = \frac{w_{y}}{w_{x}}\).
03
Substitute the components
Substitute \(w_{x}=5\) and \(w_{y}=-1\) into the formula: \(\tan(\theta) = \frac{-1}{5}\).
04
Calculate the inverse tangent
Use the arctan function to find \(\theta\): \(\theta = \tan^{-1}\left(\frac{-1}{5}\right)\). Using a calculator, \(\theta \approx -11.31^\text{掳}\).
05
Adjust for the standard position
Since the angle is measured from the positive x-axis and involves a negative y-component, \(\theta\) lies in the fourth quadrant. Convert the negative angle to standard position by adding 360掳: \(348.69^\text{掳}\).
06
Round to the nearest degree
Round the angle \(348.69^\text{掳}\) to the nearest degree. The final direction angle is \(349^\text{掳}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
vector components
A vector can be split into two parts: its x-component and y-component. These are called vector components. Think of a vector like an arrow. The x-component measures how far the arrow goes along the x-axis, and the y-component measures how far it goes along the y-axis.
In our exercise, the vector is given as \(\textbf{w}=5\textbf{i}-\textbf{j}\). This tells us that the x-component, \(w_{x}\), is 5 and the y-component, \(w_{y}\), is -1. This means the arrow goes 5 units to the right and 1 unit down.
In our exercise, the vector is given as \(\textbf{w}=5\textbf{i}-\textbf{j}\). This tells us that the x-component, \(w_{x}\), is 5 and the y-component, \(w_{y}\), is -1. This means the arrow goes 5 units to the right and 1 unit down.
tangent function
The tangent function, often written as \(\tan\), is a ratio that helps us find the angle of a right triangle. For a vector in standard position, it helps find the angle between the vector and the positive x-axis.
The tangent of an angle \(\theta\) is given by the ratio of the opposite side to the adjacent side in a right triangle. For vectors, the formula is \(\tan(\theta) = \frac{w_{y}}{w_{x}}\).
In our case, we have \(\tan(\theta) = \frac{-1}{5}\). This means the angle \(\theta\) is such that its tangent is -0.2.
The tangent of an angle \(\theta\) is given by the ratio of the opposite side to the adjacent side in a right triangle. For vectors, the formula is \(\tan(\theta) = \frac{w_{y}}{w_{x}}\).
In our case, we have \(\tan(\theta) = \frac{-1}{5}\). This means the angle \(\theta\) is such that its tangent is -0.2.
inverse tangent
To find an angle when we know its tangent, we use the inverse tangent function, written as \(\tan^{-1}\) or \(\text{arctan}\).
This function takes a ratio and gives us the corresponding angle. For example, to find the angle \(\theta\) for our vector, we use \(\theta = \tan^{-1} \bigg( \frac{-1}{5} \bigg)\).
Using a calculator, we find \(\theta \approx -11.31^\text{掳}\). This negative result means our angle is measured clockwise from the positive x-axis.
This function takes a ratio and gives us the corresponding angle. For example, to find the angle \(\theta\) for our vector, we use \(\theta = \tan^{-1} \bigg( \frac{-1}{5} \bigg)\).
Using a calculator, we find \(\theta \approx -11.31^\text{掳}\). This negative result means our angle is measured clockwise from the positive x-axis.
standard position
An angle's standard position means its initial side is along the positive x-axis, and we measure it counter-clockwise. Sometimes, we get negative angles, which are measured clockwise.
If our angle is negative, we convert it to a positive angle by adding 360掳. In our problem, \(\theta \approx -11.31^\text{掳}\) is the clockwise angle. To convert, add 360掳: \(348.69^\text{掳}\).
This tells us the angle measured counter-clockwise is \(348.69^\text{掳}\).
If our angle is negative, we convert it to a positive angle by adding 360掳. In our problem, \(\theta \approx -11.31^\text{掳}\) is the clockwise angle. To convert, add 360掳: \(348.69^\text{掳}\).
This tells us the angle measured counter-clockwise is \(348.69^\text{掳}\).
angle in fourth quadrant
When dealing with vectors, it's important to know which quadrant the angle is in. The coordinate plane is divided into four quadrants:
This helps us understand the direction of the vector and confirm that the standard position angle \(348.69^\text{掳}\) makes sense.
- First Quadrant: both x and y are positive
- Second Quadrant: x is negative, y is positive
- Third Quadrant: both x and y are negative
- Fourth Quadrant: x is positive, y is negative
This helps us understand the direction of the vector and confirm that the standard position angle \(348.69^\text{掳}\) makes sense.
