Textbook Question
Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. B = -4.0i + 4.0j
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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 11d
Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction.
Verified step by step guidance1
Step 1: Understand the vector components. The vector \( \mathbf{a} \) is given as \( \mathbf{a} = 20\mathbf{i} + 10\mathbf{j} \), where \( \mathbf{i} \) represents the x-component and \( \mathbf{j} \) represents the y-component. This means the x-component of the vector is 20 m/s², and the y-component is 10 m/s².
Step 2: Draw the vector. On a Cartesian coordinate system, plot the x-component (20 m/s²) along the positive x-axis and the y-component (10 m/s²) along the positive y-axis. The vector \( \mathbf{a} \) is represented as the diagonal of the rectangle formed by these components, starting from the origin.
Step 3: Calculate the magnitude of the vector. Use the Pythagorean theorem: \( |\mathbf{a}| = \sqrt{(a_x)^2 + (a_y)^2} \), where \( a_x = 20 \) m/s² and \( a_y = 10 \) m/s². Substitute these values into the formula to find the magnitude.
Step 4: Determine the direction of the vector. The direction is given by the angle \( \theta \) that the vector makes with the positive x-axis. Use the formula \( \theta = \tan^{-1}\left(\frac{a_y}{a_x}\right) \), where \( a_x = 20 \) m/s² and \( a_y = 10 \) m/s². Substitute these values to find the angle.
Step 5: Label the vector. On your diagram, label the vector \( \mathbf{a} \) with its magnitude and the angle \( \theta \) you calculated. Ensure the angle is measured counterclockwise from the positive x-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation
Vectors are quantities that have both magnitude and direction, represented in a coordinate system. In this case, the vector a = (20i + 10j) m/s² can be visualized in a two-dimensional plane, where 'i' represents the x-component and 'j' represents the y-component. Understanding how to graphically represent vectors is essential for visualizing their direction and magnitude.
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Magnitude of a Vector
The magnitude of a vector is a measure of its length, calculated using the Pythagorean theorem. For the vector a = (20i + 10j) m/s², the magnitude can be found using the formula |a| = √(x² + y²), where x and y are the components of the vector. This concept is crucial for determining how strong or large the vector is in physical terms.
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Direction of a Vector
The direction of a vector is specified by the angle it makes with a reference axis, typically the positive x-axis. This angle can be calculated using the tangent function, where θ = arctan(y/x). For the vector a = (20i + 10j) m/s², finding the angle helps in understanding how the vector is oriented in space, which is important for applications in physics.
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Related Practice
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
Draw each of the following vectors, label an angle that specifies the vector's direction, and then find the vector's magnitude and direction. v = (14i - 11j) m/s
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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. r = (-2.0i - 1.0j) cm
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Textbook Question
Let A = 2i + 3j, B = 2i - 4j, and C = A + B. Draw a coordinate system and on it show vectors A, B, and C.
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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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