Ch 01: Units, Physical Quantities & Vectors

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 01: Units, Physical Quantities & Vectors

Problem 31d

Chapter 1, Problem 31d

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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Textbook Question

Vector A has y-component A_y = +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

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

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 sum A + B

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Textbook Question

Find the magnitude and direction of the vector represented by the following pairs of components: A_x = −8.60 cm, A_y = 5.20 cm

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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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