Textbook Question
For the vectors A and B in Fig. E1.24, use a scale drawing to find the magnitude and direction of the vector difference A − B.
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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 28a
Young & Freedman Calc 14th EditionCh 01: Units, Physical Quantities & VectorsProblem 28aChapter 1, Problem 28a
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
Verified step by step guidance1
Start by understanding that the angle θ is the angle between the vector A and the positive x-axis. This angle can be found using trigonometric functions, specifically the tangent function, which relates the opposite side to the adjacent side in a right triangle.
The components of the vector A are given as Ax = 2.00 m and Ay = -1.00 m. These components represent the horizontal and vertical sides of the triangle formed by the vector.
Use the tangent function to find the angle θ. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, tan(θ) = Ay / Ax. Substitute the given values into this equation: tan(θ) = -1.00 m / 2.00 m.
To find the angle θ, take the arctangent (inverse tangent) of the ratio calculated in the previous step. This will give you θ = arctan(-1.00 m / 2.00 m).
Since the tangent function can have multiple solutions, consider the direction of the vector components. The negative Ay indicates that the vector is pointing downward, so the angle θ will be in the fourth quadrant. Adjust the angle accordingly to ensure it 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 Components
Vector components are the projections of a vector along the axes of a coordinate system. In this problem, Ax and Ay represent the horizontal and vertical components of vector A, respectively. Understanding these components is crucial for determining the direction and magnitude of the vector.
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Vector Addition By Components
Trigonometric Functions
Trigonometric functions, such as tangent, sine, and cosine, relate the angles and sides of a right triangle. To find the angle θ that a vector makes with the x-axis, the tangent function is used, defined as the ratio of the opposite side (Ay) to the adjacent side (Ax). This relationship helps calculate the angle from the vector components.
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Inverse Trigonometric Functions
Inverse trigonometric functions, like arctan, are used to find angles from known ratios of sides. In this context, arctan(Ay/Ax) will yield the angle θ that the vector makes with the +x-axis. This function is essential for converting the component ratio into an angle measurement.
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Related Practice
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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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
A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45° east of south, and then 280 m at 30° east of north. After a fourth displacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement.
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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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