(Effect of rotating axes on vector components) Two students, Asha and Ravi

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

(Effect of rotating axes on vector components) Two students, Asha and Ravi, describe the same point P in a plane using the same origin O but different sets of perpendicular axes. Asha uses the usual x– and y–axes, while Ravi uses x′– and y′–axes obtained by rotating Asha’s axes anticlockwise by

30^{\circ}

. In Asha’s system the coordinates of P are

(6\,\text{m},\,2\,\text{m})

. (i) Write the position vector

\mathbf r_{OP}

in Asha’s unit-vector notation. (ii) Derive expressions for the x′ and y′ components of

\mathbf r_{OP}

in Ravi’s system in terms of Asha’s components

(x, y)

and the rotation angle

\theta

. (iii) Using

\theta = 30^{\circ}

and the given values of

x

and

y

, compute the numerical values of

(x', y')

for point P in Ravi’s coordinate system. (iv) Explain why the magnitude of the position vector remains the same in both descriptions even though the individual components change.

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