Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 17a

Chapter 3, Problem 17a

Let E = 2i + 3j and F = 2i - 2j. Find the magnitude of E and F.

Verified step by step guidance

Step 1: Recall the formula for the magnitude of a vector. For a vector \( \mathbf{A} = a_i \mathbf{i} + a_j \mathbf{j} \), the magnitude \( |\mathbf{A}| \) is given by \( |\mathbf{A}| = \sqrt{a_i^2 + a_j^2} \).

Step 2: For the vector \( \mathbf{E} = 2\mathbf{i} + 3\mathbf{j} \), identify the components: \( a_i = 2 \) and \( a_j = 3 \). Substitute these values into the magnitude formula: \( |\mathbf{E}| = \sqrt{2^2 + 3^2} \).

Step 3: Simplify the expression for \( |\mathbf{E}| \): \( |\mathbf{E}| = \sqrt{4 + 9} \).

Step 4: For the vector \( \mathbf{F} = 2\mathbf{i} - 2\mathbf{j} \), identify the components: \( a_i = 2 \) and \( a_j = -2 \). Substitute these values into the magnitude formula: \( |\mathbf{F}| = \sqrt{2^2 + (-2)^2} \).

Step 5: Simplify the expression for \( |\mathbf{F}| \): \( |\mathbf{F}| = \sqrt{4 + 4} \).

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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 or size, calculated using the Pythagorean theorem. For a vector represented as E = ai + bj, the magnitude is given by |E| = √(a² + b²). This concept is essential for determining how 'strong' or 'large' a vector is in a given space.

Vector Components

Vectors can be broken down into components along the coordinate axes, typically represented as i (horizontal) and j (vertical) in two-dimensional space. For example, the vector E = 2i + 3j has a component of 2 along the x-axis and 3 along the y-axis. Understanding these components is crucial for calculating the magnitude and direction of vectors.

Vector Addition

Vector addition involves combining two or more vectors to form a resultant vector. This is done by adding their corresponding components. For instance, if E and F are added, the resultant vector R = E + F is found by summing the i components and the j components separately. This concept is fundamental in physics for analyzing forces and motion.

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Related Practice

Textbook Question

Let A = 4i - 2j, B = -3i + 5j, and F = A - 4B. What are the magnitude and direction of vector F?

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Textbook Question

Let A = 4i - 2j, B = -3i + 5j, and F = A - 4B. Draw a coordinate system and on it show vectors A, B, and F.

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Textbook Question

Let E = 2i + 3j and F = 2i - 2j. Find the magnitude of -E - 2F.

Textbook Question

FIGURE EX3.19 shows vectors A and B. What is C = A + B? Write your answer in component form using unit vectors.

Textbook Question

Let A = 4i - 2j, B = -3i + 5j, and F = A - 4B. Write vector F in component form.

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Textbook Question

What are the x- and y-components of the velocity vector shown in FIGURE EX3.20?

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