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 12a

Chapter 3, Problem 12a

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

Verified step by step guidance

Step 1: Understand the problem. You are given two vectors A and B in component form: A = 4i - 2j and B = -3i + 5j. The task is to find the resultant vector C, which is the sum of A and B, and express it in component form.

Step 2: Recall the rule for vector addition. To add two vectors in component form, you add their respective components. For example, if A = ai + bj and B = ci + dj, then A + B = (a + c)i + (b + d)j.

Step 3: Apply the rule to the given vectors. For the i-components, add the coefficients of i from A and B: 4 (from A) + (-3) (from B). For the j-components, add the coefficients of j from A and B: -2 (from A) + 5 (from B).

Step 4: Write the resultant vector C in component form. After performing the addition, C will be expressed as (sum of i-components)i + (sum of j-components)j.

Step 5: Verify your work by double-checking the addition of the components to ensure accuracy. This ensures that the resultant vector C is correctly calculated in component form.

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Key Concepts

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

Vector Addition

Vector addition involves combining two or more vectors to produce a resultant vector. This is done by adding their corresponding components. For example, if vector A has components (Ax, Ay) and vector B has components (Bx, By), the resultant vector C will have components (Ax + Bx, Ay + By).

Component Form of a Vector

The component form of a vector expresses it in terms of its horizontal and vertical components, typically represented as A = xi + yj, where x is the horizontal component and y is the vertical component. This form allows for easier calculations, especially in vector addition and subtraction.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction. The standard unit vectors in two-dimensional space are i (representing the x-direction) and j (representing the y-direction). They are essential for expressing vectors in component form and simplifying vector operations.

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Unit Vectors

Related Practice

Textbook Question

Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. B = -4.0i + 4.0j

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Textbook Question

Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction.

$\vec{a} = (20 i + 10 j) {m/s}^{2}$

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. r = (-2.0i - 1.0j) cm

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Textbook Question

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

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