Three waves of equal frequency having amplitudes 13\ , 6\ , 5\...

Let the three waves be represented by phasors $\vec{A}_1$, $\vec{A}_2$, and $\vec{A}_3$.

Taking the first wave along the x-axis, the phase angles of the three waves are $0$, $\pi/2$, and $\pi$ respectively.

The x-component of the resultant amplitude is:
$R_x = A_1 \cos(0) + A_2 \cos(\pi/2) + A_3 \cos(\pi)$
$R_x = 13(1) + 6(0) + 5(-1) = 13 - 5 = 8\ \mu\text{m}$

The y-component of the resultant amplitude is:
$R_y = A_1 \sin(0) + A_2 \sin(\pi/2) + A_3 \sin(\pi)$
$R_y = 13(0) + 6(1) + 5(0) = 6\ \mu\text{m}$

The resultant amplitude $R$ is given by:
$R = \sqrt{R_x^2 + R_y^2}$
$R = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10\ \mu\text{m}$

Answer: $10$