Table of contents

0. Math Review31m
- Math Review
  31m
1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

3. Vectors

Adding Vectors by Components

Physics

3. Vectors

Adding Vectors by Components

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6:53 minutes

Problem 24c

Textbook Question

Use components to determine the magnitude and direction of G = E＋F.

Verified step by step guidance

Break down the given vectors E and F into their components. For each vector, use the equations:

x = m a g n i t u d e \cdot \cos (θ)

for the x-component and

y = m a g n i t u d \cdot \sin (θ)

for the y-component, where

θ

is the angle of the vector with respect to the positive x-axis.

Add the x-components of vectors E and F to find the x-component of the resultant vector G:

G_{x} = E_{x} + F_{x}

Add the y-components of vectors E and F to find the y-component of the resultant vector G:

G_{y} = E_{y} + F_{y}

Determine the magnitude of the resultant vector G using the Pythagorean theorem:

G = \sqrt{{G_{x}}^{2} + {G_{y}}^{2}}

Find the direction (angle) of the resultant vector G with respect to the positive x-axis using the inverse tangent function:

θ = \tan ⁻ 1 (\frac{G_{y}}{G_{x}})

. Ensure the angle is expressed in the correct quadrant based on the signs of

G_{x}

and

G_{y}

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Key Concepts

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

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. This involves adding the corresponding components of the vectors, typically represented in a Cartesian coordinate system. The magnitude and direction of the resultant vector can be found using the Pythagorean theorem and trigonometric functions, which are essential for understanding how forces interact in physics.

Components of a Vector

Vectors can be broken down into their components along the axes of a coordinate system, usually the x and y axes. Each vector can be expressed as a sum of its horizontal (x) and vertical (y) components, allowing for easier calculations in physics. Understanding how to resolve vectors into components is crucial for accurately determining the resultant vector's magnitude and direction.

Resultant Vector

The resultant vector is the single vector that has the same effect as two or more vectors acting together. It is obtained by vector addition and represents both the total magnitude and direction of the combined vectors. In the context of forces, the resultant vector indicates the net force acting on an object, which is vital for analyzing motion and equilibrium in physics.

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Textbook Question

(II) For the vectors shown in Fig. 3–41, determine

(b) 2 A (→ above A) ― 3 B (→ above B) + 2 C (→ above C) , and

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Textbook Question

(II) Three vectors are shown in Fig. 3–41. Their magnitudes are given in arbitrary units. Determine the sum of the three vectors. Give the resultant in terms of (a) components, (b) magnitude and angle with the +𝓍 axis.

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Textbook Question

Figure 3–39 shows two vectors, $\vec{A}$ and $\vec{B}$ , whose magnitudes are A = 6.8 units and B = 5.5 units. Determine $\vec{C}$ if $\vec{C}$ = $\vec{A}$ + $\vec{B}$ . Give the magnitude and direction for each.

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Textbook Question

Let A = 4i - 2j, B = -3i + 5j, and F = A - 4B. Draw a coordinate system and on it show vectors A, B, and F.

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Textbook Question

Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction.

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Textbook Question

The minute hand on a watch is 2.0 cm in length. What is the displacement vector of the tip of the minute hand in each case? Use a coordinate system in which the y-axis points toward the 12 on the watch face. From 8:00 to 8:20 a.m.

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Textbook Question

FIGURE P3.46 shows four electric charges located at the corners of a rectangle. Like charges, you will recall, repel each other while opposite charges attract. Charge B exerts a repulsive force (directly away from B) on charge A of 3.0 N. Charge C exerts an attractive force (directly toward C) on charge A of 6.0 N. Finally, charge D exerts an attractive force of 2.0 N on charge A. Assuming that forces are vectors, what are the magnitude and direction of the net force F_netexerted on charge A?

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