Use active rotations on column vectors and positive angles counter-clockwise when viewed from the positive axis toward the origin. Rotation about

z

takes

e_{x}

(cos α, sin α, 0)

e_{y}

(- sin α, cos α, 0)

e_{z}

fixed. These images are the columns of the rotation matrix:

R_{z} (α) = cos α sin α 0 - sin α cos α 0 001 .

It satisfies

R_{z}^{T} R_{z} = I

and

det R_{z} = 1

. Stating the viewing convention avoids confusing this active rotation with a passive change of axes.

ii

Solution

With the same active rotation convention, the

x

axis remains fixed, while

e_{y}

moves toward

e_{z}

. Therefore

R_{x} (β) = 100 0 cos β sin β 0 - sin β cos β .

For instance, a positive quarter-turn maps

e_{y}

e_{z}

and

e_{z}

- e_{y}

, checking both signs. This is again an orthogonal matrix of determinant one.

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