# Thread: force vectors

1. ## force vectors

The measures of the angles between the resultant and two applied forces are 65 degrees and 42 degrees. If the magnitude of the resultant is 24 pounds, find to the nearest pound the magnitude of the smaller force.

2. sketch a parallelogram with one angle = 107 degrees

the diagonal that connects the opposite congruent 107 degree angles is the resultant 24 lb force ... the sides of the parallelogram are the two summed force vectors.

work out the values of other angles using your parallel postulates from geometry, then use the law of sines to solve for the side lengths

I found a solution.
But there may be (there should be) a more elegant one.

The measures of the angles between the resultant
. . and two applied forces are 65° and 42°.
If the magnitude of the resultant is 24 pounds,
. . find to the nearest pound the magnitude of the smaller force.
Code:
      R                       S
*                       *
|*                    * |*
| *                 *   | *
|  * b         24 *     |  *
|   *           *       |   *
|    *        *         |    *
|     * 65° *           |     *
|      *  * 42°         |      *
* - - - * - - - - - - - * - - - *
P           a           Q

The two vectors are: . $\begin{array}{ccccc}\vec u &=& \overrightarrow{PQ} &=& \langle a,\:0\rangle\qquad\qquad \\ \vec v &=& \overrightarrow{PR} &=& \langle b\cos107^o,b\sin107^o\rangle \end{array}$

. . .The resultant is: . $\vec w \;=\;\overrightarrow{PS} \;=\;\langle24\cos42^o, 24\sin42^o\rangle$

We have: . $\vec u + \vec v \:=\:\vec w \quad\Rightarrow\quad \langle a,0\rangle + \langle b\cos107^o,b\sin107^o\rangle \:=\:\langle24\cos42^o, 24\sin42^o\rangle$

Hence: . $\begin{array}{ccc}a + b\cos107^o &=& 24\cos42^o \\ b\sin107^o &=& 24\sin42^o \end{array}$

Solve the system for $a$ and $b$ and see which is smaller.