3.3 Vector addition and subtraction: analytical methods (Page 2/6)

College physics for ap® courses Page 2 / 6

Then $A = 10.3$ blocks and $θ = 29.1º$ , so that

A_{x} = A cos θ = (10.3 blocks) (cos 29.1º) = 9.0 blocks

A_{y} = A sin θ = (10.3 blocks) (sin 29.1º) = 5.0 blocks .

Calculating a resultant vector

If the perpendicular components $A_{x}$ and $A_{y}$ of a vector $A$ are known, then $A$ can also be found analytically. To find the magnitude $A$ and direction $θ$ of a vector from its perpendicular components $A_{x}$ and $A_{y}$ , we use the following relationships:

A = \sqrt{A_{x^{2}} + A_{y^{2}}}

θ = {tan}^{- 1} (A_{y} / A_{x}) .

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 $A = \sqrt{9^{2} {+5}^{2}} =10 . 3$ blocks, again consistent with the example of the person walking in a city. Finally, the direction is $θ = {tan}^{–1} (5/9) =29.1º$ , as before.

Determining vectors and vector components with analytical methods

Equations $A_{x} = A cos θ$ and $A_{y} = A sin θ$ are used to find the perpendicular components of a vector—that is, to go from $A$ and $θ$ to $A_{x}$ and $A_{y}$ . Equations $A = \sqrt{A_{x}^{2} + A_{y}^{2}}$ and $θ = {tan}^{–1} (A_{y} / A_{x})$ are used to find a vector from its perpendicular components—that is, to go from $A_{x}$ and $A_{y}$ to $A$ and $θ$ . Both processes are crucial to analytical methods of vector addition and subtraction.

Adding vectors using analytical methods

To see how to add vectors using perpendicular components, consider [link] , in which the vectors $A$ and $B$ are added to produce the resultant $R$ .

Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. — Vectors $A$ and $B$ are two legs of a walk, and $R$ is the resultant or total displacement. You can use analytical methods to determine the magnitude and direction of $R$ .

If $A$ and $B$ represent two legs of a walk (two displacements), then $R$ is the total displacement. The person taking the walk ends up at the tip of $R .$ 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, $R_{x}$ and $R_{y}$ . If we know $R_{x}$ and $R_{y}$ , we can find $R$ and $θ$ using the equations $A = \sqrt{{A_{x}}^{2} + {A_{y}}^{2}}$ and $θ = {tan}^{–1} (A_{y} / A_{x})$ . 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 $A_{x} = A cos θ$ and $A_{y} = A sin θ$ to find the components. In [link] , these components are $A_{x}$ , $A_{y}$ , $B_{x}$ , and $B_{y}$ . The angles that vectors $A$ and $B$ make with the x -axis are $θ_{A}$ and $θ_{B}$ , respectively.

Two vectors A and B are shown. The tail of the vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The horizontal and vertical components of the vectors A and B are shown with the help of dotted lines. The vectors labeled as A sub x and A sub y are the components of vector A, and B sub x and B sub y as the components of vector B.. — To add vectors $A$ and $B$ , first determine the horizontal and vertical components of each vector. These are the dotted vectors $A_{x}$ , $A_{y}$ , $B_{x}$ and $B_{y}$ shown in the image.

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] ,

R_{x} = A_{x} + B_{x}

and

R_{y} = A_{y} + B_{y} .

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Source: OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14

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