Question

What is the value of ar.ax?

(a) sin⁡θ cos⁡φ

(b) sin⁡θ sin⁡φ

(c) cos⁡θ cos⁡θ

(d) cos⁡φ sin⁡θ

I have been asked this question during an online interview.

My doubt stems from Conversion from Cartesian, Cylindrical and Spherical Coordinates in section Vector Differential Calculus of Engineering Mathematics

Answer 1

Correct option is (a) sin⁡θ cos⁡φ

To elaborate: Using the formula to convert from Spherical coordinates to Cartesian coordinates and substituting the value of the vector here in Spherical coordinates,

\(\begin{bmatrix}

Px\\

Py\\

Pz\\

\end{bmatrix} \)

\(= \begin{bmatrix}

sin\theta cos⁡φ & cos\theta cos⁡φ & -sinφ\\

sin\theta sin⁡φ & cos⁡\theta sinφ & cosφ\\

cos\theta & -sin\theta & 0\\

\end{bmatrix} \)

\(= \begin{bmatrix}

1\\

0\\

0\\

\end{bmatrix} \), and doing dot product we get, sin⁡θ cos⁡φ.