Question

Module 1- Challenge Question 1
Hi guys, I was going through the questions for module 1 and I'm not sure how they got the relations (1.4) from equation 1.3.
Any elaboration would be helpful, thank you.

Answer 1

specifically how they got the relations

0 = -3V +W + X + 2(Y + Z)
0 = V
1 = -2X

from [T] = [M*L ^{-3}]^{V} [L]^{W}[L*T ^{-2}]^{X}[L ^{2}]^{Y}[L ^{2}]^{Z}\]

Answer 2

\[\qquad\qquad [T] = [M\cdot L ^{-3}]^{V} [L]^{W}[L\cdot T ^{-2}]^{X}[L ^{2}]^{Y}[L ^{2}]^{Z}\\ ~\\
[M]^0[L]^0[T]^1 =[M]^V\cdot[L]^{-3V+W+X+2Y+2Z}[T]^{-2X}\]
the dimensions are independent so
\[
[M]^0=[M]^V\qquad\implies\qquad 0=V\\~\\
[T]^1=[T]^{-2X}\qquad\implies\qquad1=-2X\\\hspace{15em}X=-1/2\\~\\
[L]^0=[L]^{-3V+W+X+2Y+2Z}\qquad\implies\qquad 0=-3V+W+X+2Y+2Z\\\hspace{20em}0=0+W-1/2+2Y+2Z~\\\]