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The graphical method

# The graphical method - The graphical method Assume two...

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The graphical method Assume two vectors and are defined. If is added to , a third vector is created (see Figure 3.1). Figure 3.1. Commutative Law of Vector Addition. There are however two ways of combining the vectors and (see Figure 3.1). Inspection of the resulting vector shows that vector addition satisfies the commutative law (order of addition does not influence the final result): Vector addition also satisfies the associative law (the result of vector addition is independent of the order in which the vectors are added, see Figure 3.2): Figure 3.2. Associative Law of Vector Addition. Figure 3.3. Vector and - The opposite of vector is a vector with the same magnitude as but pointing in the opposite direction (see Figure 3.3): + (- ) = 0 Subtracting from is the same as adding the opposite of to (see Figure 3.4):

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= - = + (- ) Figure 3.4. Vector Subtraction. Figure 3.4 also shows that + = . In actual calculations the graphical method is not practical, and the vector algebra is performed on its components (this is the analytical method). 3.1.2. The analytical method To demonstrate the use of the analytical method of vector addition, we limit ourselves to 2 dimensions. Define a coordinate system with an x-axis and y-axis (see Figure 3.5). We can always find 2 vectors, and , whose vector sum equals . These two vectors, and , are called the components of , and by definition satisfy the following relation: Suppose that [theta] is the angle between the vector and the x-axis. The 2 components of are defined such that their direction is along the x-axis and y-axis. The length of each of the components can now be easily calculated: and the vector can be written as: Figure 3.5. Decomposition of vector . Note
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The graphical method - The graphical method Assume two...

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