<h2>Calculating a Resultant Vector</h2>
<p>If the perpendicular components \({\mathbf{A}}_{x}\) and \({\mathbf{A}}_{y}\) of a vector \(\mathbf{A}\) are known, then \(\mathbf{A}\) can also be found analytically. To find the magnitude \(A\) and direction \(\theta \) of a vector from its perpendicular components \({\mathbf{A}}_{x}\) and \({\mathbf{A}}_{y}\), we use the following relationships:</p>
<div>\(A=\sqrt{{A}_{{x}^{2}}+{A}_{{y}^{2}}}\)</div>
<div>\(\theta ={\text{tan}}^{-1}\left({A}_{y}/{A}_{x}\right)\text{.}\)</div>
<div class="wp-caption alignnone" style="width: 253px"><img alt="Vector A is shown with its horizontal and vertical components A sub x and A sub y respectively. The magnitude of vector A is equal to the square root of A sub x squared plus A sub y squared. The angle theta of the vector A with the x axis is equal to inverse tangent of A sub y over A sub x" height="350" src="https://data.ulearngo.com/assets/f9cd4c37-9106-4f87-912a-c4d85b85454b" width="253"/><p class="wp-caption-text">The magnitude and direction of the resultant vector can be determined once the horizontal and vertical components \({A}_{x}\) and \({A}_{y}\) have been determined.</p></div>
<p>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 \(\theta ={\text{tan}}^{–1}\left(\text{5/9}\right)=29.1º\), as before.</p>
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<h3 data-type="title">Determining Vectors and Vector Components with Analytical Methods</h3>
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<p>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 \(\theta ={\text{tan}}^{\text{–1}}\left({A}_{y}/{A}_{x}\right)\) 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.</p>
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Calculating a Resultant Vector

Physics is a science discipline concerned with the nature and properties of matter and energy. Physics topics include mechanics, heat, light and radiation, sound, electricity, magnetism, the structure of atoms, and so on.

Physics

Explore the principles of two-dimensional kinematics, including mastering vector addition and subtraction, resolving a vector into components, and analyzing projectile motion and relative velocities.


Calculating a Resultant Vector

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