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Note that the equation $A=\sqrt{{A}_{x}^{2}+{A}_{y}^{2}}$ is just the Pythagorean theorem relating the legs of a right triangle to the length of the hypotenuse. For example, if ${A}_{x}$ and ${A}_{y}$ are 9 and 5 blocks, respectively, then $A=\sqrt{{9}^{2}{\text{+5}}^{2}}\text{=10}\text{.}3$ blocks, again consistent with the example of the person walking in a city. Finally, the direction is $\theta ={\text{tan}}^{\mathrm{\u20131}}(\text{5/9})\mathrm{=29.1\xba}$ , as before.
Equations ${A}_{x}=A\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta $ and ${A}_{y}=A\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta $ are used to find the perpendicular components of a vector—that is, to go from $A$ and $\theta $ to ${A}_{x}$ and ${A}_{y}$ . Equations $A=\sqrt{{A}_{x}^{2}+{A}_{y}^{2}}$ and $\theta ={\text{tan}}^{\text{\u20131}}({A}_{y}/{A}_{x})$ are used to find a vector from its perpendicular components—that is, to go from ${A}_{x}$ and ${A}_{y}$ to $A$ and $\theta $ . Both processes are crucial to analytical methods of vector addition and subtraction.
To see how to add vectors using perpendicular components, consider [link] , in which the vectors $\mathbf{A}$ and $\mathbf{B}$ are added to produce the resultant $\mathbf{R}$ .
If $\mathbf{A}$ and $\mathbf{B}$ represent two legs of a walk (two displacements), then $\mathbf{R}$ is the total displacement. The person taking the walk ends up at the tip of $\mathbf{R}.$ There are many ways to arrive at the same point. In particular, the person could have walked first in the x -direction and then in the y -direction. Those paths are the x - and y -components of the resultant, ${\mathbf{R}}_{x}$ and ${\mathbf{R}}_{y}$ . If we know ${\mathbf{\text{R}}}_{x}$ and ${\mathbf{R}}_{y}$ , we can find $R$ and $\theta $ using the equations $A=\sqrt{{{A}_{x}}^{2}+{{A}_{y}}^{2}}$ and $\theta ={\text{tan}}^{\mathrm{\u20131}}({A}_{y}/{A}_{x})$ . When you use the analytical method of vector addition, you can determine the components or the magnitude and direction of a vector.
Step 1. Identify the x- and y-axes that will be used in the problem. Then, find the components of each vector to be added along the chosen perpendicular axes . Use the equations ${A}_{x}=A\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta $ and ${A}_{y}=A\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta $ to find the components. In [link] , these components are ${A}_{x}$ , ${A}_{y}$ , ${B}_{x}$ , and ${B}_{y}$ . The angles that vectors $\mathbf{A}$ and $\mathbf{B}$ make with the x -axis are ${\theta}_{\text{A}}$ and ${\theta}_{\text{B}}$ , respectively.
Step 2. Find the components of the resultant along each axis by adding the components of the individual vectors along that axis . That is, as shown in [link] ,
and
Components along the same axis, say the x -axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The same is true for components along the y -axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes it easier to add them. Now that the components of $\mathbf{R}$ are known, its magnitude and direction can be found.
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