# How to solve multiple vector simultaneous equations in closed form

I am stuck with these vectors simultaneous equations, which I got from mechanical mechanism relations:
$\\c_1=b_1-b_2+a_1-a_2$
$\\a_4+b_4=b_5+a_5$
$\\c_2=a_1+b_1-a_4-b_4$
$\\c_1/2=a_4+b_4-a_3-b_3$
$\\-c_1/2=a_5+b_5-a_3-b_3$
$\\c_2=a_2+b_2-a_5-b_5$
$\\c_1.c_2=0$ orthogonal vectors
$\\b_1$ to $\\b_5$,$\\c_1$ , $\\c_2$ are the vectors variables that we solve for, their magnitudes are known,
$\\a_1$ to $\\a_5$ are known constants , $\\b3$ y component is 0.
Are those equations enough to solve for those unknowns since we have seven unknowns in seven vector equations, that is supposed to yield 21 scalar equations. Any help is appreciated

If all of these quantities are vectors, you have a total of 19 equations, not 7 (the first 6 are 3 equations each, while the last one is really a single equation). So in terms of components you have 19 equations and 21 unknowns, so you don't have enough.

• Thanks for prompt response, I have $\\b_3$ y component is 0, does it reduce unknowns to 20? – samy Jan 30 '15 at 13:38
• Yes, unless this component is required to be zero somehow by the equations above. – Paul Jan 30 '15 at 13:43
• I will try to find the missing equation, but in the mean time, assume we have it, how to handle those equations, how to solve them together, I found myself going in a loop. – samy Jan 30 '15 at 14:06
• actually, vector $\\b_3$ has only one degree of freedom around y, meaning one direction angle in x-z plane should be enough to define it, thus the equations should be enough for the variable because variable are reduced to 19, I managed to solve for y components, but stuck with x and z components of the variables, any ideas? – samy Feb 2 '15 at 8:42