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The analytical method

# The analytical method - The analytical method To...

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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 : in contrast to Halliday, Resnick, and Walker we use and to indicate the unit vectors along the x- and y-axis, respectively (Halliday, Resnick, and Walker use i , j , and k which is harder to write). The decomposition of vector into 2 components is not unambiguous. It depends on the choice of the coordinate system (see Figure 3.6). Although the decomposition of a vector depends on the coordinate system chosen, relations between vectors are not affected by the choice of the coordinate system (for example, if two vectors are perpendicular in one coordinate system, they are perpendicular in every coordinate system). The physics (and the relation between physical quantities) is also not affected by the choice of the coordinate system, and we usually choose the coordinate system such that our problems can be solved most easily. This was already demonstrated in Chapter 2, where the origin of the coordinate system was usually defined as the position of the object at time t = 0. In

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