Textbook Question
In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?
c. no triangle
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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 5
CONCEPT PREVIEW 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.
-b
Verified step by step guidance1
Identify the vector \( \mathbf{b} \) from the given set of vectors. Understand its direction and magnitude based on the sketch or description provided.
To find \( -\mathbf{b} \), reverse the direction of vector \( \mathbf{b} \) while keeping its magnitude the same. This means if \( \mathbf{b} \) points in a certain direction, \( -\mathbf{b} \) points exactly opposite.
Draw the vector \( -\mathbf{b} \) starting from the origin or the same initial point as \( \mathbf{b} \), but pointing in the opposite direction.
Label the vector \( -\mathbf{b} \) clearly on your sketch to distinguish it from \( \mathbf{b} \).
Review your sketch to ensure that the length of \( -\mathbf{b} \) matches that of \( \mathbf{b} \) and that the direction is exactly reversed.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Addition and Subtraction
Vector addition involves combining two vectors by placing them head-to-tail and drawing the resultant vector from the start of the first to the end of the second. Subtraction, such as -b, means reversing the direction of vector b before adding it. Understanding how to add and subtract vectors graphically is essential for solving problems involving vector sums.
Recommended video:
05:29
Adding Vectors Geometrically
Parallelogram Rule
The parallelogram rule is a geometric method to find the resultant of two vectors originating from the same point. By placing the vectors tail-to-tail, you complete a parallelogram with these vectors as adjacent sides; the diagonal of this parallelogram represents their sum. This rule helps visualize vector addition and is crucial for accurate vector sketching.
Recommended video:
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Vector Direction and Negative Vectors
A negative vector has the same magnitude as the original but points in the opposite direction. For example, -b is vector b reversed. Recognizing how to represent negative vectors graphically is important for correctly performing vector subtraction and understanding vector operations in trigonometry.
Recommended video:
05:13Finding Direction of a Vector
Related Practice
Textbook Question
CONCEPT PREVIEW 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.
2c
Textbook Question
Solve each triangle. See Examples 2 and 3.
C = 45.6°, b = 8.94 m, a = 7.23 m
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Textbook Question
Determine the number of triangles ABC possible with the given parts.
a = 50, b = 26, A = 95°
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Textbook Question
Use the law of sines to find the indicated part of each triangle ABC.
Find b if a = 165 m, A = 100.2°, B = 25.0°
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Textbook Question
CONCEPT PREVIEW Assume a triangle ABC has standard labeling.
a. Determine whether SAA, ASA, SSA, SAS, or SSS is given.
b. Determine whether the law of sines or the law of cosines should be used to begin solving the triangle.
a, B, and C