Cylindrical Coordinates — Question 7

Question 7 – Solution

Strategy. Expand $(x_1-x_2)^2+(y_1-y_2)^2$ and use the cosine difference identity.

Step 1: Formula For cylindrical points, $$d^2=r_1^2+r_2^2-2r_1r_2\cos(\theta_1-\theta_2)+(z_1-z_2)^2.$$ This follows from the planar law of cosines plus the vertical difference.

Step 2: Substitute Here $\theta_1-\theta_2=-2\pi/3$, so its cosine is $-1/2$. Thus $$\begin{align*} d^2&=2^2+5^2-2(2)(5)(-1/2)+(1-4)^2\\&=4+25+10+9=48. \end{align*}$$ Therefore $\boxed{d=4\sqrt 3}$.

Step 3: Check The horizontal squared distance is $4+25+10=39$; adding vertical squared distance $9$ gives $48$, consistent with perpendicular horizontal and vertical components.