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Chapter 6: Problem 98

Find the magnitude and direction angle for the vector \(\langle-3,-9\rangle\)

Short Answer

Expert verified
Magnitude: \(3\sqrt{10}\); Direction Angle: \(251.57^\circ\)

Step by step solution

01

Find the Magnitude

To find the magnitude of the vector \(\langle-3,-9\rangle\), use the formula \[ \|\mathbf{v}\| = \sqrt{x^2 + y^2} \]. Substitute \( x = -3 \) and \( y = -9 \) into the formula: \[ \|\mathbf{v}\| = \sqrt{(-3)^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10} \]
02

Calculate the Direction Angle

To determine the direction angle \( \theta \) of the vector, use the formula \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]. Substitute \( x = -3\) and \( y = -9 \) into the formula: \[ \theta = \tan^{-1}\left(\frac{-9}{-3}\right) = \tan^{-1}(3) \]. The inverse tangent function yields an angle in the fourth or first quadrant, but since both components are negative, the vector is in the third quadrant: \[ \theta = 180^\circ + \tan^{-1}(3) \]
03

Compute and Adjust the Result

Using a calculator, find \[ \tan^{-1}(3) \approx 71.57^\circ \]. Add this to \( 180^\circ \) to find the final direction angle: \[ 180^\circ + 71.57^\circ = 251.57^\circ \]

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Key Concepts

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

magnitude calculation
The first step in understanding vectors is to calculate their magnitude. The magnitude essentially represents how long the vector is. To compute the magnitude, you use the formula:
direction angle
Once you know the magnitude, the next step is calculating the direction angle of the vector. This angle tells you the orientation of the vector in the coordinate system. To find the direction angle, we begin with the formula: This yields the initial angle with reference to the standard position, which we'll adjust later.
inverse tangent function
Next, we use the inverse tangent function to find the initial angle. This function, denoted as , finds the angle whose tangent is a given value. After inputting our values, we find the following result:
third quadrant
Finally, to correctly identify the vector's direction, we need to consider its position within the coordinate plane. Since both vector components are negative, our vector lies in the third quadrant. Therefore, we adjust the angle accordingly:

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Most popular questions from this chapter

Graph each pair of polar equations on the same screen of your calculator and use the trace feature to estimate the polar coordinates of all points of intersection of the curves. Check your calculator manual to see how to graph polar equations. $$ r=3 \sin 4 \theta, r=2 $$

Write each expression in the form \(a+\) bi where \(a\) and \(b\) are real numbers. $$ (4-5 i)^{2} $$

For each rectangular equation, write an equivalent polar equation. $$ (x+1)^{2}+y^{2}=4 $$

An equilateral triangle and a regular hexagon have equal perimeters. What is the ratio of the area of the triangle to the area of the hexagon?

Write each expression in the form \(a+\) bi where \(a\) and \(b\) are real numbers. $$ i^{34}+i^{9} $$
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