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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 29d
Capitolo 1, Problema 29d

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


Figure E1.24 shows vectors A (8.00m), B (15.0m), C (12.0m) with angles in a coordinate system.

Guida verificata passo dopo passo
1
Identify the components of vector A. Since 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 B - A. Subtract the components of A from B: (B - A)_x = B_x - A_x and (B - A)_y = B_y - A_y.
Determine the magnitude of the vector difference B - A using the Pythagorean theorem: |B - A| = sqrt((B - A)_x^2 + (B - A)_y^2).
Find the direction of the vector difference B - A by calculating the angle θ with respect to the positive x-axis using the tangent function: θ = arctan((B - A)_y / (B - A)_x).

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Vector Components

Vectors can be broken down into their components along the x and y axes. This is done using trigonometric functions: the x-component is found using the cosine of the angle, while the y-component is found using the sine. For example, for a vector A at an angle θ, the components are A_x = A * cos(θ) and A_y = A * sin(θ). Understanding components is essential for performing vector 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 achieved by adding the negative of the vector being subtracted. For vectors A and B, the difference B - A can be calculated by determining the components of both vectors and then subtracting the corresponding components: (B_x - A_x, B_y - A_y). This method allows for a clear understanding of the resultant vector's magnitude and direction.
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Subtracting Vectors Graphically

Magnitude and Direction of Vectors

The magnitude of a vector is its length, calculated using the Pythagorean theorem for its components: |V| = √(V_x² + V_y²). The direction is typically expressed as an angle relative to a reference axis, often using the arctangent function: θ = arctan(V_y / V_x). Knowing how to find both the magnitude and direction is crucial for fully describing a vector in a two-dimensional space.
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Calculating Magnitude & Components of a Vector
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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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