Textbook Question
Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈15, -8〉
Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 10
Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈-4, -7〉
Verified step by step guidance1
Identify the components of the vector. Here, the vector is given as \(\langle -4, -7 \rangle\), where \(x = -4\) and \(y = -7\).
Calculate the magnitude of the vector using the formula \(\text{magnitude} = \sqrt{x^2 + y^2}\). Substitute the values to get \(\sqrt{(-4)^2 + (-7)^2}\).
Find the direction angle \(\theta\) of the vector relative to the positive x-axis using the formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Substitute the values to get \(\theta = \tan^{-1}\left(\frac{-7}{-4}\right)\).
Since both \(x\) and \(y\) are negative, the vector lies in the third quadrant. Adjust the angle found in the previous step by adding \(180^\circ\) to get the correct direction angle.
Round the direction angle to the nearest tenth of a degree as required.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Magnitude
The magnitude of a vector 〈x, y〉 represents its length and is calculated using the Pythagorean theorem as √(x² + y²). This gives a non-negative scalar value indicating how long the vector is regardless of its direction.
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Finding Magnitude of a Vector
Direction Angle of a Vector
The direction angle of a vector is the angle it makes with the positive x-axis, measured counterclockwise. It can be found using the arctangent function: θ = arctan(y/x), with adjustments based on the vector's quadrant to get the correct angle.
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Quadrant Considerations for Angles
Since arctan(y/x) only returns values between -90° and 90°, the vector's quadrant must be considered to determine the correct direction angle. For vectors in the third quadrant (both x and y negative), add 180° to the arctan value to find the proper angle.
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Related Practice
Textbook Question
Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈5, 7〉
Textbook Question
Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.
a + b
Textbook Question
Consider each case and determine whether there is sufficient information to solve the triangle using the law of sines.
Three sides are known.
1
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Textbook Question
Find each angle B. Do not use a calculator.
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4
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Textbook Question
Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.
〈-7, 24〉
3
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