Question 980592
Draw the vectors where the tails are at the same point. The vectors are shown in blue. The resultant vector is shown in red.



Also, add in auxiliary lines to form a parallelogram. These are the black dashed lines.



Because we have a parallelogram, the adjacent angles are supplementary. So 100+x = 180 means x = 80 degrees. Also, the opposite sides of any parallelogram are congruent. All of this is shown below in the drawing.



<img src = "http://i150.photobucket.com/albums/s91/jim_thompson5910/7-21-2015%204-14-28%20PM_zpschsqcw3f.png">



The goal is to find the length of the red vector. To do this, we use the law of cosines {{{c^2 = a^2 + b^2 - 2*a*b*cos(C)}}}



In this case, the sides of the triangle are {{{a = 3}}}, {{{b = 4}}}, and 'c' is unknown. Angle C is 80 degrees. Solve for c to get...



{{{c^2 = a^2 + b^2 - 2*a*b*cos(C)}}}



{{{c^2 = 3^2 + 4^2 - 2*3*4*cos(80)}}}



{{{c^2 = 9 + 16 - 24*cos(80)}}}



{{{c^2 = 25 - 24*cos(80)}}}



{{{c^2 = 25 - 24*0.173648}}}



{{{c^2 = 25 - 4.167552}}}



{{{c^2 = 20.832448}}}



{{{sqrt(c^2) = sqrt(20.832448)}}}



{{{c = 4.564258}}}



The magnitude of the resultant is approximately <font color="red">4.564258 units</font> long.