For the vectors shown in Fig. 3–41, determine 2 A - 3 B + 2 C.

Name: Physics for Scientists and Engineers
Author: Giancoli Douglas
ISBN: 9780137488179

Verified step by step guidance

Step 1: Understand the problem. You are tasked with finding the resultant vector from the expression 2A - 3B + 2C, where A, B, and C are vectors. Refer to Fig. 3–41 for the components or directions of these vectors.

Step 2: Break each vector (A, B, and C) into its components. For example, if vector A has components Ax and Ay, write A as A = (Ax, Ay). Do the same for vectors B and C.

Step 3: Multiply each vector by its respective scalar. For instance, multiply vector A by 2 to get 2A = (2Ax, 2Ay). Similarly, calculate -3B = (-3Bx, -3By) and 2C = (2Cx, 2Cy).

Step 4: Add the components of the scaled vectors together. For the x-components, compute: (2Ax) + (-3Bx) + (2Cx). For the y-components, compute: (2Ay) + (-3By) + (2Cy).

Step 5: Combine the results from Step 4 to express the resultant vector in component form as R = (Rx, Ry), where Rx is the sum of the x-components and Ry is the sum of the y-components. If needed, you can also find the magnitude and direction of the resultant vector using the formulas: |R| = sqrt(Rx^2 + Ry^2) and θ = arctan(Ry/Rx).

Key Concepts

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

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. This involves adding the corresponding components of the vectors, which can be visualized graphically by placing the tail of one vector at the head of another. The resultant vector represents the cumulative effect of the individual vectors in both magnitude and direction.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For example, multiplying a vector by 2 doubles its length, while multiplying by -1 reverses its direction. This concept is essential when manipulating vectors in expressions like 2A or -3B.

Linear Combination of Vectors

A linear combination of vectors involves creating a new vector by multiplying each vector by a scalar and then adding the results. In the expression 2A - 3B + 2C, each vector A, B, and C is scaled by its respective coefficient, and the resulting vectors are summed. This concept is fundamental in vector spaces and is used to express complex vector relationships.

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