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 34

Chapter 8, Problem 34

Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
v - u

Verified step by step guidance

Identify the given vectors: \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \).

Recall that vector subtraction \( \mathbf{v} - \mathbf{u} \) is performed component-wise: subtract the corresponding components of \( \mathbf{u} \) from \( \mathbf{v} \).

Set up the subtraction for each component: \( (v_x - u_x, v_y - u_y) \), where \( v_x = 4, u_x = -2, v_y = 3, u_y = 5 \).

Calculate each component difference separately: \( 4 - (-2) \) for the x-component and \( 3 - 5 \) for the y-component.

Combine the results to write the vector \( \mathbf{v} - \mathbf{u} \) as \( \langle 4 - (-2), 3 - 5 \rangle \).

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Key Concepts

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

Vector Representation

Vectors are quantities defined by both magnitude and direction, often represented as ordered pairs or tuples in two dimensions. For example, u = 〈-2, 5〉 indicates a vector with components -2 along the x-axis and 5 along the y-axis.

Vector Subtraction

Vector subtraction involves subtracting corresponding components of two vectors. For vectors v = 〈v₁, v₂〉 and u = 〈u₁, u₂〉, the difference v - u is 〈v₁ - u₁, v₂ - u₂〉, resulting in a new vector.

Component-wise Operations

Operations on vectors such as addition and subtraction are performed component-wise, meaning each component is handled independently. This simplifies calculations and helps visualize vector operations geometrically.

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Related Practice

Textbook Question

A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N 38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse?

Textbook Question

Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.

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Textbook Question

To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582 yd. She then finds that T = 32° 50' and R = 102° 20'. Find RS. See the figure.

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Textbook Question

Apply the law of sines to the following:

A = 104°, a = 26.8, b = 31.3.

What happens when we try to find the measure of angle B using a calculator?

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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

Without using the law of sines, explain why no triangle ABC can exist that satisfies A = 103° 20', a = 14.6 ft, b = 20.4 ft.

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