The vectors a and b represent two forces acting at the same point, and θ is the smallest positiv

Name: Problem 40, Chapter 7: Precalculus: Functions and Graphs
Uploaded: 2021-06-01T18:58:35-08:00
Duration: 55 s
Channel: Wendi Zhao
Description: The vectors a and b represent two forces acting at the same point, and θ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force. 𝐚=30 lb, 𝐛=50 lb, θ=150^∘

step 1 As shown in the figure, we let A in the positive direction of the x -axis, and then we decompose

Question

The vectors a and b represent two forces acting at the same point, and $\boldsymbol{\theta}$ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force. $$\|\mathbf{a}\|=30 \mathrm{lb}, \quad\|\mathbf{b}\|=50 \mathrm{lb}, \quad \theta=150^{\circ}$$

   The vectors a and b represent two forces acting at the same point, and $\boldsymbol{\theta}$ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force.
$$\|\mathbf{a}\|=30 \mathrm{lb}, \quad\|\mathbf{b}\|=50 \mathrm{lb}, \quad \theta=150^{\circ}$$

Precalculus: Functions and Graphs

Earl W. Swokowski,… 12th Edition

Chapter 7, Problem 40

Instant Answer

Solved by Expert Wendi Zhao

06/01/2021

Step 1

The horizontal component of $\mathbf{b}$ is $\|\mathbf{b}\|\cos(\theta)$ and the vertical component is $\|\mathbf{b}\|\sin(\theta)$. Show more…

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The vectors a and b represent two forces acting at the same point, and $\boldsymbol{\theta}$ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force. $$\|\mathbf{a}\|=30 \mathrm{lb}, \quad\|\mathbf{b}\|=50 \mathrm{lb}, \quad \theta=150^{\circ}$$

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Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. In the context of forces, this means that the overall effect (resultant force) is determined by both the magnitudes and the directions of the individual forces acting at a point.

Resultant Force Magnitude

The magnitude of the resultant force when two vectors are added is determined by both their individual magnitudes and the angle between them. This concept is crucial in understanding how forces combine to produce a net effect on an object.

Law of Cosines

The law of cosines is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used in vector addition to calculate the magnitude of the resultant vector when the vectors are not perpendicular. The formula allows the computation of the resultant force by incorporating the angle between the two force vectors.

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