Ch. 7 - Applications of Trigonometry and Vectors

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 7 - Applications of Trigonometry and Vectors

Problem 31c

Chapter 8, Problem 31c

Use the figure to find each vector: - u. Use vector notation as in Example 4.

Verified step by step guidance

Identify the vector \( \mathbf{u} \) from the figure, noting its direction and magnitude or its components if given.

Recall that the vector \( -\mathbf{u} \) is the vector \( \mathbf{u} \) reversed in direction but with the same magnitude.

If \( \mathbf{u} \) is given in component form as \( \mathbf{u} = \langle x, y \rangle \), then \( -\mathbf{u} = \langle -x, -y \rangle \).

If the vector \( \mathbf{u} \) is given graphically, determine its components by measuring or using trigonometric relationships based on the angle and length.

Write the vector \( -\mathbf{u} \) explicitly in vector notation, ensuring the direction is opposite to \( \mathbf{u} \) and the magnitude remains the same.

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Key Concepts

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

Vector Notation

Vector notation represents vectors using components along coordinate axes, typically written as ⟨x, y⟩ in two dimensions. This notation simplifies vector operations like addition, subtraction, and scalar multiplication by expressing vectors as ordered pairs or triples.

Vector Direction and Magnitude

A vector has both magnitude (length) and direction. Understanding how to determine these from a figure is essential, as the vector's components correspond to its horizontal and vertical displacements, which define its direction and size.

Vector Operations (Negation)

Negating a vector reverses its direction while keeping its magnitude the same. If vector u = ⟨x, y⟩, then -u = ⟨-x, -y⟩. This concept is crucial when the question asks for -u, indicating the vector pointing opposite to u.

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Related Practice

Textbook Question

Apply the law of sines to the following: a = √5, c = 2√5, A = 30°. What is the value of sin C? What is the measure of C? Based on its angle measures, what kind of triangle is triangle ABC?

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Textbook Question

Use the figure to find each vector: u - v. Use vector notation as in Example 4.

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Textbook Question

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

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Textbook Question

Solve each triangle. See Examples 2 and 3.

A = 112.8°, b = 6.28 m, c = 12.2 m

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Textbook Question

Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.

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Textbook Question

Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5°. Find the direction and magnitude of the equilibrant.

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