Expand $\left(\frac{x}{2}+\frac{2 y}{3}-\frac{3 z}{4}\right)^{2}$

To expand the expression $\left(\frac{x}{2}+\frac{2 y}{3}-\frac{3 z}{4}\right)^{2}$,we use the algebraic identity $(a+b+c)^{2} = a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$.
Here,$a = \frac{x}{2}$,$b = \frac{2y}{3}$,and $c = -\frac{3z}{4}$.
Substituting these values into the identity:
$= \left(\frac{x}{2}\right)^{2} + \left(\frac{2y}{3}\right)^{2} + \left(-\frac{3z}{4}\right)^{2} + 2\left(\frac{x}{2}\right)\left(\frac{2y}{3}\right) + 2\left(\frac{2y}{3}\right)\left(-\frac{3z}{4}\right) + 2\left(-\frac{3z}{4}\right)\left(\frac{x}{2}\right)$
$= \frac{x^{2}}{4} + \frac{4y^{2}}{9} + \frac{9z^{2}}{16} + \frac{2xy}{3} - yz - \frac{3zx}{4}$