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Chapter 8: Problem 17

Find the magnitude and direction of the vector $$ \langle 2,-5\rangle $$

Short Answer

Expert verified
Magnitude: \( \sqrt{29} \approx 5.39 \), Direction: \( -68.20^\circ \).

Step by step solution

01

Identify the Components of the Vector

The vector given is \( \langle 2, -5 \rangle \). This means that the vector has a horizontal component \( x = 2 \) and a vertical component \( y = -5 \).
02

Calculate the Magnitude of the Vector

The magnitude of a vector \( \langle a, b \rangle \) is calculated using the formula: \( \sqrt{a^2 + b^2} \). For our vector \( \langle 2, -5 \rangle \), substitute \( a = 2 \) and \( b = -5 \):\[magnitude = \sqrt{2^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}\approx 5.39\]
03

Find the Direction (Angle) of the Vector

The direction of the vector \( \langle a, b \rangle \) is given by the angle \( \theta \), which can be found using:\[ \theta = \arctan\left(\frac{b}{a}\right) \]Substitute \( a = 2 \) and \( b = -5 \):\[\theta = \arctan\left(\frac{-5}{2}\right) \approx -68.20^\circ\]Since the vector lies in the fourth quadrant (positive \( x \), negative \( y \)), the angle is measured clockwise from the positive x-axis.

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Key Concepts

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

Vector Components
Vectors are entities in mathematics and physics characterized by a magnitude and direction. Each vector can be broken down into horizontal and vertical parts, known as vector components. The vector \( \langle 2, -5 \rangle \) means that the horizontal component (often called the x-component) is 2, and the vertical component (the y-component) is -5. Consider these components as coordinate points that, when connected, form the vector. Connect a point at (0,0) to (2,-5) in a graph, you see the vector's path.
  • The horizontal component tells you how far along the x-axis to move: 2 units right.
  • The vertical component tells you how far along the y-axis to move: 5 units down.
Recognizing components helps in visualizing the vector and is essential for further calculations.
Magnitude Calculation
The magnitude of a vector is a measure of its length, answering how far apart the vector鈥檚 start and end points are. To calculate the magnitude, use Pythagoras鈥� Theorem on the components of the vector. For a vector \( \langle a, b \rangle \), the formula is:\[ magnitude = \sqrt{a^2 + b^2} \]Plug in the components of our vector \( \langle 2, -5 \rangle \):
  • Square each component: \(2^2 = 4\), \((-5)^2 = 25\).
  • Add them together: \(4 + 25 = 29\).
  • Find the square root: \(\sqrt{29} \approx 5.39\).
This result, approximately 5.39, is the vector鈥檚 magnitude, representing its length in the space of two dimensions.
Vector Direction
Vector direction specifies the angle at which the vector is pointing relative to a reference axis, typically the positive x-axis. For the vector \( \langle 2, -5 \rangle \), we find this direction by determining the angle of orientation or direction angle. The angle can be found by looking at which quadrant the vector is located in:
  • If both components are positive, it lies in the first quadrant.
  • If x is negative and y is positive, it lies in the second quadrant.
  • If both components are negative, it's in the third quadrant.
  • If x is positive and y is negative, it is in the fourth quadrant.
For \( \langle 2, -5 \rangle \), since \( x = 2 \) and \( y = -5 \), the vector lies in the fourth quadrant. This direction is measured clockwise, not the standard counterclockwise, because the y-component is negative.
Arctangent Function
The arctangent function, denoted as \( \arctan \), is used to determine the angle of a vector based on its components. For any vector \( \langle a, b \rangle \), the direction or angle \( \theta \) can be calculated by:\[ \theta = \arctan\left(\frac{b}{a}\right) \]In the example \( \langle 2, -5 \rangle \), this calculation becomes:
  • \( \frac{-5}{2} = -2.5 \)
  • \( \theta = \arctan(-2.5) \approx -68.20^\circ \)
Since this angle is measured clockwise from the positive x-axis, it provides the direction uniquely for vectors pointing downward. The arctangent function is fundamental when determining vector directions, especially in fields like physics where angle measurement is crucial.

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