Textbook Question
In each case, find the x- and y- components of vector A: A = 11.2j - 9.91i
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Ch 01: Units, Physical Quantities & Vectors
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors37
- Ch 02: Motion Along a Straight Line57
- Ch 03: Motion in Two or Three Dimensions45
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws54
- Ch 06: Work & Kinetic Energy11
- Ch 07: Potential Energy & Conservation6
- Ch 08: Momentum, Impulse, and Collisions15
- Ch 09: Rotation of Rigid Bodies37
- Ch 10: Dynamics of Rotational Motion44
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics27
- Ch 13: Gravitation34
- Ch 14: Periodic Motion26
- Ch 15: Mechanical Waves22
- Ch 16: Sound & Hearing28
- Ch 17: Temperature and Heat28
- Ch 18: Thermal Properties of Matter35
- Ch 19: The First Law of Thermodynamics29
- Ch 20: The Second Law of Thermodynamics18
- Ch 21: Electric Charge and Electric Field28
- Ch 22: Gauss' Law21
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF24
- Ch 26: Direct-Current Circuits29
- Ch 27: Magnetic Field and Magnetic Forces16
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction22
- Ch 30: Inductance14
- Ch 31: Alternating Current20
- Ch 33: The Nature and Propagation of Light17
- Ch 34: Geometric Optics29
- Ch 35: Interference13
- Ch 36: Diffraction19
- Ch 37: Special Relativity19
- Ch 38: Photons: Light Waves Behaving as Particles12
- Ch 39: Particles Behaving as Waves25
- Ch 40: Quantum Mechanics I: Wave Functions20
- Ch 41: Quantum Mechanics II: Atomic Structure21
- Ch 42: Molecules and Condensed Matter17
- Ch 43: Nuclear Physics13
- Ch 44: Particle Physics and Cosmology17
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
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Young & Freedman Calc 15th EditionCh 01: Units, Physical Quantities & VectorsProblem 30a
Young & Freedman Calc 15th EditionCh 01: Units, Physical Quantities & VectorsProblem 30aChapter 1, Problem 30a
Find the magnitude and direction of the vector represented by the following pairs of components: Ax = −8.60 cm, Ay = 5.20 cm
Verified step by step guidance1
Step 1: Understand that the vector components Ax and Ay represent the horizontal and vertical components of the vector, respectively. The magnitude of the vector can be found using the Pythagorean theorem.
Step 2: Use the formula for the magnitude of a vector: , where A and B are the components Ax and Ay. Substitute Ax = -8.60 cm and Ay = 5.20 cm into the formula.
Step 3: Calculate the magnitude of the vector using the formula: cm.
Step 4: Determine the direction of the vector using the tangent function. The direction angle θ can be found using the formula: . Substitute Ax = -8.60 cm and Ay = 5.20 cm into the formula.
Step 5: Calculate the direction angle θ using the formula: . Consider the signs of Ax and Ay to determine the correct quadrant for the angle.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Magnitude
The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector with components Ax and Ay, the magnitude is given by the formula √(Ax² + Ay²). This provides a scalar quantity representing the size of the vector irrespective of its direction.
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Vector Direction
The direction of a vector is determined by the angle it makes with a reference axis, typically the positive x-axis. This angle, θ, can be found using trigonometry, specifically the tangent function: θ = arctan(Ay/Ax). The angle provides insight into the vector's orientation in a coordinate system.
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Coordinate System
Understanding the coordinate system is crucial for interpreting vector components. In a Cartesian coordinate system, vectors are expressed in terms of their horizontal (x-axis) and vertical (y-axis) components. This system allows for the decomposition and analysis of vectors in two dimensions, facilitating calculations of magnitude and direction.
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Related Practice
Textbook Question
Vector A is 2.80 cm long and is 60.0° above the x-axis in the first quadrant. Vector B is 1.90 cm long and is 60.0° below the x-axis in the fourth quadrant (Fig. E1.35). Use components to find the magnitude and direction of A - B In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.
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Textbook Question
A disoriented physics professor drives 3.25 km north, then 2.20 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.
Textbook Question
For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of the vector difference B - A
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Textbook Question
For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of the vector sum A + B
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Textbook Question
For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of the vector difference A - B
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