Ch 01: Units, Physical Quantities & Vectors

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 01: Units, Physical Quantities & Vectors

Problem 36a

Chapter 1, Problem 36a

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

Verified step by step guidance

To find the magnitude of a vector, use the formula: \( |\mathbf{A}| = \sqrt{A_x^2 + A_y^2} \), where \( A_x \) and \( A_y \) are the components of vector \( \mathbf{A} \).

For vector \( \mathbf{A} = 4\mathbf{i} + 7\mathbf{j} \), identify the components: \( A_x = 4 \) and \( A_y = 7 \).

Substitute the components of vector \( \mathbf{A} \) into the magnitude formula: \( |\mathbf{A}| = \sqrt{4^2 + 7^2} \).

Similarly, for vector \( \mathbf{B} = 5\mathbf{i} - 2\mathbf{j} \), identify the components: \( B_x = 5 \) and \( B_y = -2 \).

Substitute the components of vector \( \mathbf{B} \) into the magnitude formula: \( |\mathbf{B}| = \sqrt{5^2 + (-2)^2} \).

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Key Concepts

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

Vector Magnitude

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector A = ai + bj, the magnitude is given by |A| = √(a² + b²). This formula helps determine the size of the vector in the coordinate plane.

Vector Components

Vectors are represented by components along the coordinate axes, typically denoted as i and j for the x and y axes, respectively. Understanding vector components is crucial for calculating vector magnitude and performing operations like addition or subtraction.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, which is essential for calculating vector magnitudes.

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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

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

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Textbook Question

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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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

Given two vectors A = -2i + 3j + 4k and B = 3.00î + 1.00ĵ − 3.00k, find the magnitude of each vector.

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Textbook Question

Find the scalar product of the vectors A and B given in Exercise 1.38.

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