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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 38
Chapter 8, Problem 38

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

Verified step by step guidance
1
Identify the given vectors: \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \).
Understand that scalar multiplication means multiplying each component of the vector by the scalar. For example, \( 2\mathbf{u} = \langle 2 \times (-2), 2 \times 5 \rangle \).
Calculate \( 2\mathbf{u} \) by multiplying each component of \( \mathbf{u} \) by 2: \( 2\mathbf{u} = \langle -4, 10 \rangle \).
Calculate \( 4\mathbf{v} \) by multiplying each component of \( \mathbf{v} \) by 4: \( 4\mathbf{v} = \langle 16, 12 \rangle \).
Add the resulting vectors component-wise: \( 2\mathbf{u} + 4\mathbf{v} = \langle -4 + 16, 10 + 12 \rangle \).

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Key Concepts

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

Vector Addition and Scalar Multiplication

Vector addition involves adding corresponding components of two vectors to form a new vector. Scalar multiplication means multiplying each component of a vector by a scalar (a real number). These operations allow combining and scaling vectors, essential for expressions like 2u + 4v.
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Component-wise Operations

Vectors in two dimensions are represented by ordered pairs. Operations such as addition and scalar multiplication are performed component-wise, meaning each x-component and y-component is handled separately. This simplifies calculations and helps visualize vector results.
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Notation and Vector Representation

Vectors are often denoted by angle brackets, e.g., 〈x, y〉, representing their components along the x and y axes. Understanding this notation is crucial for interpreting and manipulating vectors in problems involving vector arithmetic.
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Related Practice
Textbook Question

The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.

Textbook Question

To build the pyramids in Egypt, it is believed that giant causeways were constructed to transport the building materials to the site. One such causeway is said to have been 3000 ft long, with a slope of about 2.3°. How much force would be required to hold a 60-ton monolith on this causeway?


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Textbook Question

A balloonist is directly above a straight road 1.5 mi long that joins two villages. She finds that the town closer to her is at an angle of depression of 35°, and the farther town is at an angle of depression of 31°. How high above the ground is the balloon? 


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Textbook Question

A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizontal? 

Textbook Question

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?

Textbook Question

Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal.