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Magazine Chapter 10: Problem 47
\(47-52\) . Find the magnitude and direction (in degrees) of the vector. $$ \mathbf{v}=\langle 3,4\rangle $$
Short Answer
Expert verified
Magnitude is 5, direction is approximately 53.13 degrees.
Step by step solution
01
Understanding the Vector Components
First, notice that the vector \( \mathbf{v} = \langle 3, 4 \rangle \) has components \(3\) and \(4\), where \(3\) is the horizontal component (along the x-axis) and \(4\) is the vertical component (along the y-axis).
02
Calculating the Magnitude
To find the magnitude (length) of the vector \( \mathbf{v} \), we use the formula for the Euclidean norm: \( \| \mathbf{v} \| = \sqrt{3^2 + 4^2}.\) Compute this to find \( \| \mathbf{v} \| = \sqrt{9 + 16} = \sqrt{25} = 5 \). Thus, the magnitude of the vector is \(5\).
03
Determining the Direction
The direction of a vector is typically calculated using the angle \( \theta \) between the vector and the positive x-axis. Use the tangent function for this, \( \tan \theta = \frac{{4}}{{3}} \). Use the arctangent function to find \( \theta = \tan^{-1} \left( \frac{4}{3} \right) \), which is approximately \(53.13^\circ\).
04
Final Step: Verify Direction and Description
Since both components \(3\) and \(4\) are positive, the vector is in the first quadrant. Therefore, the angle \(53.13^\circ\) with the positive x-axis, and upwards from the horizontal, correctly represents the direction of the vector.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Components
Vectors are often represented as a pair of numbers in a form like \( \mathbf{v} = \langle 3, 4 \rangle \). Here, the components \(3\) and \(4\) signify the direction and magnitude along the x-axis and y-axis, respectively. These are the horizontal and vertical parts that make up the vector.
- The horizontal component \(3\) shows how far along the x-axis the vector extends.
- The vertical component \(4\) tells us how far along the y-axis it reaches.
Euclidean Norm
The Euclidean norm, or magnitude, of a vector is akin to measuring the length of a vector from its tail to its head. For a vector \( \mathbf{v} = \langle x, y \rangle \), the Euclidean norm can be found using the formula \( \| \mathbf{v} \| = \sqrt{x^2 + y^2} \).
For our vector \( \langle 3, 4 \rangle \), the computation is straightforward:
For our vector \( \langle 3, 4 \rangle \), the computation is straightforward:
- Square the horizontal component: \(3^2 = 9\).
- Square the vertical component: \(4^2 = 16\).
- Add these squares: \(9 + 16 = 25\).
- Take the square root: \(\sqrt{25} = 5\).
Tangent Function
The tangent function is crucial when dealing with vectors and their directions. It relates the opposite and adjacent sides of a right triangle formed by the vector components.
For a vector \( \langle x, y \rangle \), the tangent of the angle \( \theta \) between the vector and the positive x-axis is given by \( \tan \theta = \frac{y}{x} \). In our scenario, the vector \( \langle 3, 4 \rangle \) forms a right triangle:
For a vector \( \langle x, y \rangle \), the tangent of the angle \( \theta \) between the vector and the positive x-axis is given by \( \tan \theta = \frac{y}{x} \). In our scenario, the vector \( \langle 3, 4 \rangle \) forms a right triangle:
- Opposite side (y-component): 4 units.
- Adjacent side (x-component): 3 units.
Arctangent
The arctangent, sometimes marked as \( \tan^{-1} \), is the inverse function of tangent. It is used to find the angle whose tangent is a given number.
To find the angle \(\theta\) for the vector \( \langle 3, 4 \rangle \), where \( \tan \theta = \frac{4}{3} \), we calculate \( \theta = \tan^{-1} \left( \frac{4}{3} \right) \). This process yields an angle of approximately \(53.13^\circ\).
To find the angle \(\theta\) for the vector \( \langle 3, 4 \rangle \), where \( \tan \theta = \frac{4}{3} \), we calculate \( \theta = \tan^{-1} \left( \frac{4}{3} \right) \). This process yields an angle of approximately \(53.13^\circ\).
- Step 1: Calculate \( \frac{4}{3} \).
- Step 2: Apply the arctangent function.
- Result: \( \theta = 53.13^\circ \).
