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 31c

Chapter 1, Problem 31c

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

Verified step by step guidance

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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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, 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.

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.

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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Calculating Magnitude & Components of a Vector

Related Practice

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

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: A_x = 2.00m, A_y = −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 sum A + B

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