Problem 71
Question
Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=4 \mathbf{i}-2 \mathbf{j}\)
Step-by-Step Solution
Verified Answer
-26.57°
1Step 1: Identify the components of the vector
Given vector \(\frac{4}{3}\textbf{i} - \frac{2}{3}\textbf{j} \), identify the components \(a\) and \(b\). Here, \(a = 4\) and \(b = -2\).
2Step 2: Set up the formula for the direction angle
The direction angle \( \theta \) of a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \) is given by \( \theta = \tan^{-1}\left( \frac{b}{a} \right) \).
3Step 3: Substitute the components into the formula
Substitute \(a = 4\) and \(b = -2\) into the formula: \( \theta = \tan^{-1}\left( \frac{-2}{4} \right) \).
4Step 4: Calculate the arctangent
Calculate \( \tan^{-1}\left( -\frac{1}{2} \right) \). This gives \( \theta \approx -26.57^{\circ} \).
5Step 5: Adjust the angle to the correct quadrant
Since \(a = 4 > 0\) and \(b = -2 < 0\), the vector is in the fourth quadrant. Thus, the direction angle \( \theta \) is \( -26.57^{\circ} \).
Key Concepts
Finding direction anglesArctangent functionComponents of a vectorVector quadrants
Finding direction angles
In mathematics and physics, the direction angle of a vector is a key concept that describes the orientation of the vector in relation to a specified coordinate system. To find the direction angle of a vector \textbf{v}, we generally follow a series of steps. These steps involve identifying the vector's components, applying a formula, and ensuring the angle is in the correct quadrant. The direction angle \( \theta \) is typically measured clockwise or counterclockwise from the positive x-axis. Remember, this is very useful for navigation, physics problems, and engineering applications.
To illustrate, consider a vector \( \mathbf{v} = 4 \, \mathbf{i} - 2 \, \mathbf{j} \). The direction angle tells us how far this vector bends from the x-axis.
To illustrate, consider a vector \( \mathbf{v} = 4 \, \mathbf{i} - 2 \, \mathbf{j} \). The direction angle tells us how far this vector bends from the x-axis.
Arctangent function
The arctangent function, denoted as \( \tan^{-1} \) or \( \, \text{atan} \, \), is crucial when finding direction angles. This function helps us determine the angle given the ratio of the opposite side to the adjacent side in a right triangle. In the context of vectors, it provides the angle from the ratio of the vector's y-component to its x-component.
For example, if we have a vector \( \mathbf{v} = a \, \mathbf{i} + b \, \mathbf{j} \), the direction angle \( \theta \) can be found using the formula:
\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]
When using this formula, we must be mindful that the value from the arctangent function gives us an angle which might be in the incorrect quadrant, so we adjust it accordingly based on the vector’s components.
For example, if we have a vector \( \mathbf{v} = a \, \mathbf{i} + b \, \mathbf{j} \), the direction angle \( \theta \) can be found using the formula:
\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]
When using this formula, we must be mindful that the value from the arctangent function gives us an angle which might be in the incorrect quadrant, so we adjust it accordingly based on the vector’s components.
Components of a vector
Vectors are defined by their magnitude and direction. The components of a vector break it into parts along the x- and y-axes. In a 2-dimensional plane, any vector \( \textbf{v} \) can be expressed in terms of its components as: \( \mathbf{v} = a \, \mathbf{i} + b \, \mathbf{j} \), where \( a \) and \( b \) are the scalar coefficients for the unit vectors \( \mathbf{i} \) (x-axis) and \( \mathbf{j} \) (y-axis), respectively.
Identifying these components is the first step toward finding the direction angle. For example, for a vector given as \( 4 \, \mathbf{i} - 2 \, \mathbf{j} \), we have \( a = 4 \) and \( b = -2 \). These values are then used in the arctangent function to find the direction angle.
Identifying these components is the first step toward finding the direction angle. For example, for a vector given as \( 4 \, \mathbf{i} - 2 \, \mathbf{j} \), we have \( a = 4 \) and \( b = -2 \). These values are then used in the arctangent function to find the direction angle.
Vector quadrants
Vectors lie in different quadrants of the Cartesian coordinate system based on the signs of their components. The quadrants are determined as follows:
- First Quadrant: \( a > 0 \) and \( b > 0 \)
- Second Quadrant: \( a < 0 \) and \( b > 0 \)
- Third Quadrant: \( a < 0 \) and \( b < 0 \)
- Fourth Quadrant: \( a > 0 \) and \( b < 0 \)
Other exercises in this chapter
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