If the projections of the line segment $\overline{AB}$ on the $yz-$plane, $zx-$plane, $xy-$ plane are  $\sqrt{160},\sqrt{153}\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}5$ respectively, then the projection of $\overline{AB}$on the $z-$axis is

The projection of the line segment $\overline{AB}$ on the $yz-$plane is  $\sqrt{{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}=\sqrt{160}$

The projection of the line segment $\overline{AB}$on the $zx-$plane is  $\sqrt{{\left(z_2-z_1\right)}^2+{\left(x_2-x_1\right)}^2}=\sqrt{153}$

The projection of the line segment $\overline{AB}$on the $xy-$plane is $\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}=5$

It implies that 

${\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2=169$

Hence,${\left(z_2-z_1\right)}^2=169-25=144$

Apply square root on both sides$\left|z_2-z_1\right|=\boxed{12}$ , this is the length of the projection of$\overline{AB}$  on $z-$ axis.