b
\([ {{v_c}} ] = [ {{\eta ^x}{\rho ^y}{r^z}}] \)( {given} 
Writing the dimensions of various quantities in
eqn. (\(i\)), we get
\(\left[ {{M^0}L{T^{ - 1}}} \right] = {\left[ {M{L^{ - 1}}{T^{ - 1}}} \right]^x}{\left[ {M{L^{ - 3}}{T^0}} \right]^y}{\left[ {{M^0}L{T^0}} \right]^z}\)
\( = \left[ {{M^{x + y}}{L^{ - x - 3y + z}}{T^{ - x}}} \right]\)
Applying the principle of homogeneity of

dimensions we get

\(,x + y = 0; - x - 3y + z = 1; - x =  - 1\)
On solving we get
\(x = 1,y =  - 1,z =  - 1\)