4.1. SYSTEMS OF EQUATIONS, ALGEBRAIC PROCEDURES 71

where the units on ρV 2AB are

kgm3

( msec

)2m2 =

kg×msec2

which are the units of force. Each of these gi is of the form

Ax1Bx2θx3V x4V x5

0 ρx6 µ

x7 (4.11)

and each gi is independent of the dimensions. That is, this expression must not depend onmeters, kilograms, seconds, etc. Thus, placing in the units for each of these quantities, oneneeds

mx1mx2(mx4 sec−x4

)(mx5 sec−x5

)(kgm−3)x6 (kgsec−1 m−1)x7 = m0kg0 sec0

Notice that there are no units on θ because it is just the radian measure of an angle. Henceits dimensions consist of length divided by length, thus it is dimensionless. Then this leadsto the following equations for the xi.

m : x1 + x2 + x4 + x5−3x6− x7 = 0sec : −x4− x5− x7 = 0kg : x6 + x7 = 0

Then the augmented matrix for this system of equations is 1 1 0 1 1 −3 −1 00 0 0 1 1 0 1 00 0 0 0 0 1 1 0

The row reduced echelon form is then 1 1 0 0 0 0 1 0

0 0 0 1 1 0 1 00 0 0 0 0 1 1 0

and so the solutions are of the form

x1 =−x2− x7, x3 = x3,x4 =−x5− x7,x6 =−x7

Thus, in terms of vectors, the solution is(x1 x2 x3 x4 x5 x6 x7

)=

(−x2− x7 x2 x3 −x5− x7 x5 −x7 x7

)Thus the free variables are x2,x3,x5,x7. By assigning values to these, we can obtain di-mensionless variables by placing the values obtained for the xi in the formula 4.11. Forexample, let x2 = 1 and all the rest of the free variables are 0. This yields

x1 =−1,x2 = 1,x3 = 0,x4 = 0,x5 = 0,x6 = 0,x7 = 0.