Textbook Question
Vector A has y-component Ay = +9.60 m. A makes an angle of 32.0° counterclockwise from the +y-axis. (a) What is the x-component of A? (b) What is the magnitude of A?
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Ch 01: Units, Physical Quantities & Vectors
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
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Young & Freedman Calc 14th EditionCh 01: Units, Physical Quantities & VectorsProblem 31a
Young & Freedman Calc 14th EditionCh 01: Units, Physical Quantities & VectorsProblem 31aChapter 1, Problem 31a
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
Verified step by step guidance1
Identify the components of vector A. Since vector A is along the negative y-axis, its components are: \( A_x = 0 \) and \( A_y = -8.0 \text{ m} \).
Determine the components of vector B. Vector B makes a 30° angle with the positive y-axis. The components are: \( B_x = 15.0 \text{ m} \times \sin(30°) \) and \( B_y = 15.0 \text{ m} \times \cos(30°) \).
Calculate the x and y components of vector B using trigonometric functions: \( B_x = 15.0 \text{ m} \times 0.5 \) and \( B_y = 15.0 \text{ m} \times \sqrt{3}/2 \).
Add the components of vectors A and B to find the components of the resultant vector \( \vec{R} = \vec{A} + \vec{B} \). Thus, \( R_x = A_x + B_x \) and \( R_y = A_y + B_y \).
Calculate the magnitude of the resultant vector \( \vec{R} \) using the Pythagorean theorem: \( R = \sqrt{R_x^2 + R_y^2} \). Determine the direction by finding the angle \( \theta \) with respect to the x-axis using \( \theta = \tan^{-1}(R_y/R_x) \).
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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 determine a resultant vector. This can be done graphically or analytically. In the graphical method, vectors are placed head-to-tail, while in the analytical method, the components of each vector are summed separately in the x and y directions to find the resultant's magnitude and direction.
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Vector Addition By Components
Components of a Vector
A vector can be broken down into its components along the x and y axes. The x-component is found using the cosine of the angle, while the y-component is determined using the sine of the angle. This decomposition simplifies calculations, allowing for the use of basic algebra to find the resultant vector's magnitude and direction.
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Magnitude and Direction
The magnitude of a vector is its length, representing the quantity it describes, while the direction indicates where the vector points. The magnitude can be calculated using the Pythagorean theorem when the components are known, and the direction can be found using the arctangent function to determine the angle relative to a reference axis.
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Related Practice
Textbook Question
Compute the x- and y-components of the vectors A, B, C, and D in Fig. E1.24.
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Textbook Question
Let θ be the angle that the vector A makes with the +x-axis, measured counterclockwise from that axis. Find angle θ for a vector that has these components: Ax = 2.00m, Ay = −1.00 m
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Textbook Question
A postal employee drives a delivery truck over the route shown in Fig. E1.25. Use the method of components to determine the magnitude and direction of her resultant displacement. 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.
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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 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 difference A - B
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