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Chapter 1: Problem 11

What is "vector addition" and how is it done?

Short Answer

Expert verified
Answer: The resultant vector, C, is (4, 6).

Step by step solution

01

1. Understanding Vectors

A vector is a mathematical object that has both a magnitude and a direction. Vectors can represent various kinds of physical quantities such as displacement, velocity, and force. Vectors are typically represented by arrows, where the length of the arrow represents the magnitude and the orientation of the arrow represents the direction.
02

2. Vector Addition Definition

Vector addition is the operation of combining two or more vectors to produce a new vector, called the "sum" or "resultant" vector. The main principle is that vectors can be added geometrically by connecting them head-to-tail.
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3. Geometric Method of Vector Addition

To add vectors geometrically, follow these steps: 1. Draw the first vector as an arrow, with the tail at the origin and the head pointing towards its direction. 2. Draw the second vector as an arrow with its tail at the head of the first vector and its head pointing towards its direction. 3. Draw the resultant vector from the tail of the first vector to the head of the second vector. This new vector represents the sum of the two vectors. If you have more than two vectors, repeat this process for all vectors you want to add. Note: Vector addition is commutative, meaning the order in which you add the vectors does not change the resultant vector. So A + B = B + A.
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4. Algebraic Method of Vector Addition

To perform vector addition algebraically, you need to express each vector in terms of its components. Components are projections of the vector along the coordinate axes. For example, a vector in a 2D space has two components: one along the x-axis (horizontal) and another along the y-axis (vertical). To add vectors algebraically: 1. Write down each vector in terms of its components: A = (A_x, A_y) and B = (B_x, B_y). 2. Add the corresponding components of each vector: C_x = A_x + B_x and C_y = A_y + B_y. 3. Combine the components to form the resultant vector: C = (C_x, C_y)
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5. Example of Vector Addition

Let's add two 2D vectors A = (3, 4) and B = (1, 2) using both methods. Geometric method: 1. Draw A as an arrow starting at the origin with a horizontal component of 3 units and a vertical component of 4 units. 2. Starting at the head of A, draw B with a horizontal component of 1 unit and a vertical component of 2 units. 3. Draw the resultant vector C, from the tail of A to the head of B. Algebraic method: 1. A = (3, 4), B = (1, 2) 2. C_x = 3 + 1 = 4, C_y = 4 + 2 = 6 3. C = (4, 6) In both methods, the resultant vector C has a horizontal component of 4 units and a vertical component of 6 units.

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