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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 9
Chapter 8, Problem 9

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.
Blue vectors labeled a through h are shown, with a red vector illustrating the sum of vectors a and e using the parallelogram rule.


a + b

Verified step by step guidance
1
Begin by sketching vectors \( \mathbf{a} \) and \( \mathbf{b} \) on the same coordinate plane, ensuring each vector is drawn with its initial point at the origin or a common point for clarity.
Place the initial point of vector \( \mathbf{b} \) at the terminal point of vector \( \mathbf{a} \) to prepare for vector addition using the head-to-tail method.
Draw the resultant vector \( \mathbf{a} + \mathbf{b} \) starting from the initial point of \( \mathbf{a} \) to the terminal point of \( \mathbf{b} \) after it has been moved.
Alternatively, use the parallelogram rule: place both vectors \( \mathbf{a} \) and \( \mathbf{b} \) so their initial points coincide, then complete the parallelogram formed by these two vectors.
The diagonal of the parallelogram starting from the common initial point represents the vector sum \( \mathbf{a} + \mathbf{b} \). Sketch this diagonal to visualize the resultant vector.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Addition

Vector addition involves combining two or more vectors to find a resultant vector. This is done by placing the initial point of one vector at the terminal point of another and then drawing the vector from the start of the first to the end of the last. The resultant vector represents the combined effect of the original vectors.
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Adding Vectors Geometrically

Parallelogram Rule

The parallelogram rule is a geometric method to add two vectors. By placing the vectors so their tails coincide, a parallelogram is formed using the vectors as adjacent sides. The diagonal of this parallelogram from the common tail point represents the sum of the two vectors.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Vector Representation and Sketching

Accurately sketching vectors involves drawing arrows with correct direction and relative magnitude. This visual representation helps in understanding vector operations like addition. Labeling vectors clearly and maintaining scale aids in applying rules such as the parallelogram method effectively.
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Multiplying Vectors By Scalars
Related Practice
Textbook Question

Find the length of the remaining side of each triangle. Do not use a calculator.


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Textbook Question

To find the distance AB across a river, a surveyor laid off a distance BC = 354 m on one side of the river. It is found that B = 112° 10' and C = 15° 20'. Find AB. See the figure.


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Textbook Question

Determine the number of triangles ABC possible with the given parts.


c = 50, b = 61, C = 58°

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Textbook Question

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈5, 7〉

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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.

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Textbook Question

Solve each triangle ABC that exists.

A = 96.80°, b = 3.589 ft, a = 5.818 ft

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