11.3 Vector addition and subtraction: analytical methods  (Page 2/5)

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 $\theta ={\text{tan}}^{\mathrm{–1}}(\text{5/9})\mathrm{=29.1º}$ , as before.

Adding vectors using analytical methods

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 $\theta ={\text{tan}}^{\mathrm{–1}}(A_y/A_x)$ . 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 $A_x=A\text{cos}\theta$ and $A_y=A\text{sin}\theta$ 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.