Ch. 7 - Applications of Trigonometry and Vectors

Problem 7.5

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Chapter 8, Problem 7.5

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.

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-b

Verified step by step guidance

Identify the vectors labeled as 'a' and 'e' from the provided diagram or description.

Sketch the vectors 'a' and 'e' on a coordinate plane, ensuring that their initial points coincide.

Apply the parallelogram rule: Draw a parallelogram where 'a' and 'e' serve as adjacent sides. The diagonal of this parallelogram starting from the common initial point will represent the resultant vector.

Complete the parallelogram by drawing lines parallel to vector 'a' and vector 'e' at the terminal points of each vector.

The resultant vector, which is the diagonal of the parallelogram, represents the vector sum 'a + e'. Label this vector appropriately on your sketch.

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 determine a resultant vector. This can be done graphically by placing the tail of one vector at the head of another, or by using the parallelogram rule, where two vectors are represented as adjacent sides of a parallelogram, and the diagonal represents the resultant.

Negative Vectors

A negative vector, such as -b, is a vector that has the same magnitude as vector b but points in the opposite direction. This concept is crucial for understanding how to manipulate vectors in various operations, including addition and subtraction, as it affects the resultant vector's direction.

Parallelogram Rule

The parallelogram rule is a method for finding the resultant of two vectors. By drawing a parallelogram where the two vectors are adjacent sides, the diagonal from the common initial point to the opposite corner represents the resultant vector. This rule is essential for visualizing vector addition in a two-dimensional space.

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One boat pulls a barge with a force of 100 newtons. Another boat pulls the barge at an angle of 45° to the first force, with a force of 200 newtons. Find the resultant force acting on the barge, to the nearest unit, and the angle between the resultant and the first boat, to the nearest tenth.

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

a = 50, b = 26, A = 95°

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