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Ch 01: Units, Physical Quantities & Vectors
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 15th EditionCh 01: Units, Physical Quantities & VectorsProblema 29c
Capitolo 1, Problema 29c

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


Vector diagram E1.24 with vectors A, B, C, D, and angles for vector addition.

Guida verificata passo dopo passo
1
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 m.
Identify the components of vector B. Use trigonometry to find: B_x = 15.0 m * cos(30°) and B_y = 15.0 m * sin(30°).
Calculate the components of the vector difference A - B. Subtract the components of B from A: (A - B)_x = A_x - B_x and (A - B)_y = A_y - B_y.
Determine the magnitude of the vector difference A - B using the Pythagorean theorem: |A - B| = sqrt((A - B)_x^2 + (A - B)_y^2).
Find the direction of the vector difference A - B by calculating the angle θ with respect to the x-axis using the tangent function: θ = atan((A - B)_y / (A - B)_x).

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Vector Components

Vector components are the projections of a vector along the axes of a coordinate system, typically the x and y axes. For any vector, these components can be calculated using trigonometric functions: the x-component is found using the cosine of the angle, while the y-component uses the sine. This breakdown allows for easier manipulation and analysis of vectors, especially when performing operations like addition or subtraction.
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Vector Addition By Components

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which can be visualized as adding a negative vector. Mathematically, to find the vector difference A - B, you can subtract the components of vector B from those of vector A. This results in a new vector that represents the direction and magnitude of the difference, which can then be expressed in terms of its own components.
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Subtracting Vectors Graphically

Magnitude and Direction

The magnitude of a vector is its length, representing the quantity it describes, while the direction indicates the orientation of the vector in space. To find the magnitude of a resultant vector from its components, the Pythagorean theorem is used. The direction can be determined using the arctangent function, which relates the components to the angle the vector makes with a reference axis, typically the x-axis.
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