In terms of the

coordinates of a fixed space frame $\{$s$\},$ the frame $\{$a$\}$ has its

$-$axis pointing in the direction $(0,0,1)$ and its

$-$axis pointing in the direction $(-1,0,0),$ and the frame b has its

$-$axis pointing in the direction $(1,0,0)$ and its

$-$axis pointing in the direction $(0,0,-1).$ Submit a Word document in which you answer the following questions:

Draw by hand the three frames, at different locations so that they are easy to see.

Write down the rotation matrices

and

$.$

Given

$,$ how do you calculate

without using the matrix inverse? Write down

and verify its correctness using your drawing.

Given

and

$,$ how do you calculate

without using matrix inverses? Compute the answer and verify its correctness using your drawing.

Let

be considered as a transformation operator consisting of a rotation about

by $-90\backslash$deg $.$ Calculate

and think of

as a representation of an orientation, R as a rotation of

$,$ and

as the new orientation after the rotation has been performed. Does the new orientation

correspond to a rotation of

by $-90\backslash$deg about the world$-$fixed

$-$axis or about the body$-$fixed

$-$axis? Now calculate

$.$ Does the new orientation

correspond to a rotation of

by $-90\backslash$deg about the world$-$fixed

$-$axis or about the body$-$fixed

$-$axis?

Use

to change the representation of the point pb $=(1,2,3)($which is in $\{$b$\}$ coordinates$)$ to $\{$s$\}$ coordinates.

Choose a point

$=(1,2,3)$ represented by in $\{$s$\}$ coordinates. Calculate

and

$.$ For each operation, should the result be interpreted as changing coordinates $($from the $\{$s$\}$ frame to $\{$b$\})$ without moving the point p or as moving the location of the point without changing the reference frame of the representation?

Calculate the matrix exponential corresponding to the exponential coordinates of rotation

$.$ Draw the corresponding frame relative to $\{$s$\},$ as well as the rotation axis

$.$