Calculating a Resultant Vector

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:

\(A=\sqrt{{A}_{{x}^{2}}+{A}_{{y}^{2}}}\)

\(\theta ={\text{tan}}^{-1}\left({A}_{y}/{A}_{x}\right)\text{.}\)

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

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.

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.

Determining Vectors and Vector Components with Analytical Methods

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.

This lesson is part of:

Two-Dimensional Kinematics

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