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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Ch 03: Vectors and Coordinate Systems
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 3, Problem 13b
Let A = 2i + 3j, B = 2i - 4j, and C = A + B. Draw a coordinate system and on it show vectors A, B, and C.
Verified step by step guidance1
Step 1: Understand the problem. You are given two vectors A and B in component form, and you need to find their sum (vector C). Additionally, you are asked to represent these vectors graphically on a coordinate system.
Step 2: Add the components of vectors A and B to find vector C. Use the formula for vector addition: \( C = A + B \). For the x-components, add \( 2 + 2 \), and for the y-components, add \( 3 - 4 \). This will give the components of vector C.
Step 3: Write the resulting vector C in component form. After performing the addition, vector C will be expressed as \( C = (x_{C})i + (y_{C})j \), where \( x_{C} \) and \( y_{C} \) are the calculated x and y components.
Step 4: Draw a coordinate system. Label the axes as x and y. Plot vector A starting from the origin, with its x-component (2) and y-component (3). Then plot vector B starting from the origin, with its x-component (2) and y-component (-4). Finally, plot vector C starting from the origin, using the components calculated in Step 3.
Step 5: Use arrows to represent the vectors on the graph. Ensure that the direction and magnitude of each vector are accurately represented. Label each vector (A, B, and C) clearly on the graph to complete the visualization.
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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 involves combining two or more vectors to form a resultant vector. This is done by adding their corresponding components. For example, if vector A has components (2, 3) and vector B has components (2, -4), their sum, vector C, is obtained by adding the i-components and the j-components separately, resulting in C = (2+2)i + (3-4)j = 4i - 1j.
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Coordinate System
A coordinate system is a two-dimensional plane defined by two perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical). Each point in this system can be represented by an ordered pair (x, y), where x indicates the position along the x-axis and y indicates the position along the y-axis. This system is essential for graphically representing vectors and their components.
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Vector Representation
Vectors can be represented graphically as arrows in a coordinate system, where the length of the arrow indicates the magnitude and the direction of the arrow indicates the vector's direction. For instance, vector A = 2i + 3j would be represented as an arrow starting from the origin (0,0) and pointing to the point (2,3) in the coordinate system. This visual representation helps in understanding the relationship between different vectors.
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Related Practice
Textbook Question
Let A = 4i - 2j, B = -3i + 5j, and E = 2A + 3B. What are the magnitude and direction of vector E?
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Textbook Question
Let A = 4i - 2j, B = -3i + 5j, and D = A - B. Draw a coordinate system and on it show vectors A, B, and D.
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Textbook Question
Let A = 4i - 2j, B = -3i + 5j, and C = A + B. What are the magnitude and direction of vector C?
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Textbook Question
Let A = 4i - 2j, B = -3i + 5j, and F = A - 4B. Write vector F in component form.
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Textbook Question
Let A = 4i - 2j, B = -3i + 5j, and C = A + B. Write vector C in component form.
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