Textbook Question
Solve each triangle ABC that exists.
A = 42.5°, a = 15.6 ft, b = 8.14 ft
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 21
For each pair of vectors u and v with angle θ between them, sketch the resultant.
|u| = 20, |v| = 30, θ = 30°
Verified step by step guidance1
Identify the given information: the magnitudes of vectors \( \mathbf{u} \) and \( \mathbf{v} \) are \( |\mathbf{u}| = 20 \) and \( |\mathbf{v}| = 30 \), and the angle between them is \( \theta = 30^\circ \).
Recall that the resultant vector \( \mathbf{r} = \mathbf{u} + \mathbf{v} \) can be found using the law of cosines for vectors, where the magnitude of \( \mathbf{r} \) is given by:
\[ \\|\mathbf{r}\\\| = \sqrt{ |\mathbf{u}|^2 + |\mathbf{v}|^2 + 2 |\mathbf{u}| |\mathbf{v}| \cos(\theta) } \]
To sketch the resultant, start by drawing vector \( \mathbf{u} \) as an arrow of length 20 units in any direction.
Next, from the tip of \( \mathbf{u} \), draw vector \( \mathbf{v} \) at an angle of \( 30^\circ \) relative to \( \mathbf{u} \), with length 30 units.
The resultant vector \( \mathbf{r} \) is represented by the arrow drawn from the tail of \( \mathbf{u} \) to the tip of \( \mathbf{v} \). This completes the vector addition and visually shows the resultant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Magnitude and Direction
A vector is defined by its magnitude (length) and direction. Understanding how to represent vectors graphically involves drawing arrows with lengths proportional to their magnitudes and angles corresponding to their directions. This is essential for accurately sketching vectors u and v with given magnitudes and the angle θ between them.
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Finding Components from Direction and Magnitude
Angle Between Vectors
The angle θ between two vectors is the measure of the smallest rotation from one vector to the other. It determines how the vectors are oriented relative to each other and affects the resultant vector's magnitude and direction when the vectors are added.
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04:33Find the Angle Between Vectors
Vector Addition and Resultant Vector
The resultant vector is found by adding two vectors head-to-tail, combining their magnitudes and directions. Using the law of cosines or parallelogram method, the magnitude and direction of the resultant can be determined, which is crucial for sketching the resultant vector from u and v with a known angle θ.
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05:29Adding Vectors Geometrically
Related Practice
Textbook Question
Solve each triangle ABC that exists.
B = 72.2°, b = 78.3 m, c = 145 m
Textbook Question
For each pair of vectors u and v with angle θ between them, sketch the resultant.
|u| = 12, |v| = 20, θ = 27°
Textbook Question
Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.
Textbook Question
Solve each triangle ABC.
B = 38° 40', a = 19.7 cm, C = 91° 40'
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Textbook Question
Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.