Question

Let there be a vector X = yz^2 ax + zx^2 ay + xy^2 az. Find X at P(3,6,9) in cylindrical coordinates.

(a) 100 ax – 398 ay + 108 az

(b) 103 ax – 401 ay + 109 az

(c) 105 ax – 393 ay + 105 az

(d) 95 ax – 395 ay + 100 az

The question was posed to me in unit test.

I'd like to ask this question from Conversion from Cartesian, Cylindrical and Spherical Coordinates in section Vector Differential Calculus of Engineering Mathematics

Answer 1

The correct choice is (a) 100 ax – 398 ay + 108 az

Explanation: The formula for converting a vector from Cartesian coordinates to Cylindrical coordinates is  \(\begin{bmatrix}

X\rho\\

Xφ\\

Xx\\

\end{bmatrix} \)

\(= \begin{bmatrix}

cos⁡φ & sin⁡φ & 0\\

-sin⁡φ & cos⁡φ & 0\\

0 & 0 & 1\\

\end{bmatrix} \)

\(= \begin{bmatrix}

Xx\\

Xy\\

Xz\\

\end{bmatrix} \)

Substituting the column matrix in right hand side by the given the vector, and solving the matrix we get a vector in cylindrical coordinates. Now change the point P which is in Cartesian coordinates to Cylindrical coordinates. Now, we should substitute the point P in X thus finding the value of X at P. Hence we get the value 100 ax – 398 ay + 108 az.