The vectors a and b represent two forces acting at the same point, and θis the smallest positive

Name: Problem 42, Chapter 7: Precalculus: Functions and Graphs
Uploaded: 2021-06-01T16:43:47-08:00
Duration: 1 min 48 s
Channel: Wendi Zhao
Description: The vectors a and b represent two forces acting at the same point, and θis the smallest positive angle between a and b. Approximate the magnitude of the resultant force. (a) 20 lb, S 17^∘ W (b) 50 lb, N 82^∘ W

step 1 As shown in the graph, first we decompose A and B into their vertical and horizontal components.

Question

The vectors a and b represent two forces acting at the same point, and $\theta$ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force. $$\text { (a) } 20 \mathrm{lb}, \quad \mathrm{S} 17^{\circ} \mathrm{W} \quad \text { (b) } 50 \mathrm{lb}, \quad \mathrm{N} 82^{\circ} \mathrm{W}$$

   The vectors a and b represent two forces acting at the same point, and $\theta$ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force.
$$\text { (a) } 20 \mathrm{lb}, \quad \mathrm{S} 17^{\circ} \mathrm{W} \quad \text { (b) } 50 \mathrm{lb}, \quad \mathrm{N} 82^{\circ} \mathrm{W}$$

Precalculus: Functions and Graphs

Earl W. Swokowski,… 11th Edition

Chapter 7, Problem 42

Instant Answer

Solved by Expert Wendi Zhao

06/01/2021

Step 1

The horizontal component is given by the magnitude of the vector times the cosine of the angle, and the vertical component is given by the magnitude of the vector times the sine of the angle. For vector a, the horizontal component is $20 \cos(17^\circ) = 18.7885$ Show more…

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The vectors a and b represent two forces acting at the same point, and $\theta$ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force. $$\text { (a) } 20 \mathrm{lb}, \quad \mathrm{S} 17^{\circ} \mathrm{W} \quad \text { (b) } 50 \mathrm{lb}, \quad \mathrm{N} 82^{\circ} \mathrm{W}$$

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Key Concepts

Vector Representation

Vectors are mathematical entities that represent both magnitude and direction. In physics, forces are modeled as vectors because their effects depend not only on how strong they are, but also on the direction in which they are applied.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. When multiple forces act on a point, their combined effect is found by summing these vectors, typically by adding corresponding components or using geometric methods.

Law of Cosines

The Law of Cosines is used to calculate the magnitude of the resultant vector when two vectors with known magnitudes and the angle between them are being added. This method generalizes the Pythagorean theorem and is essential for resolving scenarios where the vectors do not intersect perpendicularly.

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