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 35b

Chapter 1, Problem 35b

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.

Verified step by step guidance

First, resolve Vector A into its components. The x-component of Vector A is given by A_x = A * cos(60°) and the y-component by A_y = A * sin(60°). Substitute A = 2.80 cm to find A_x and A_y.

Next, resolve Vector B into its components. The x-component of Vector B is B_x = B * cos(60°) and the y-component is B_y = -B * sin(60°) because it is below the x-axis. Substitute B = 1.90 cm to find B_x and B_y.

To find the components of A - B, subtract the components of B from A. So, (A - B)_x = A_x - B_x and (A - B)_y = A_y - B_y.

Calculate the magnitude of the resultant vector A - B using the Pythagorean theorem: |A - B| = sqrt((A - B)_x^2 + (A - B)_y^2).

Determine the direction of the resultant vector A - B by calculating the angle θ with respect to the x-axis using the tangent function: θ = arctan((A - B)_y / (A - B)_x).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 axes of a coordinate system. For a vector at an angle, the horizontal (x) and vertical (y) components can be calculated using trigonometric functions: the x-component is found using cosine, and the y-component using sine. This breakdown is essential for performing vector addition or subtraction, as it allows for the manipulation of each component separately.

Vector Subtraction

Vector subtraction involves finding the resultant vector when one vector is taken away from another. This can be visualized as adding a negative vector, which is the same as reversing the direction of the vector being subtracted. Mathematically, if A and B are vectors, then A - B can be computed by subtracting the components of B from those of A, resulting in a new vector that represents the difference.

Magnitude and Direction

The magnitude of a vector is its length, while the direction indicates the angle it makes with a reference axis. 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. Understanding both magnitude and direction is crucial for accurately representing and interpreting vector quantities.

Recommended video:

Guided course

03:59

Calculating Magnitude & Components of a Vector

Related Practice

Textbook Question

You are given two vectors A = -3i + 6j and B = 7i + 2j. Let Counterclockwise angles be positive. What angle does A make with the +x-axis?

Textbook Question

Given two vectors A = 4i + 7j and B = 5i - 2j, find the magnitude of each vector.

views

Textbook Question

In each case, find the x- and y- components of vector A: A = 11.2j - 9.91i

views

rank

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.

views

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

views