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**Confused77** 1)a)What us the matrix, M that transforms i->u, j->v and k->w? and what is its determinant?

1)b)This matrix then transforms the cube with corners (0,0,0), (1,0,0),(0,1,0), (0,0,1), (1,0,1), (0,1,1), (1,1,0) and (1,1,1) what are the new corners and the new cubes volume?

Am I right in thinking that the matrix is [1,0,0;0,1,0;0,0,1] and that it will have no effect on the cube as this is the identity matrix? You haven't told us what u, v and w are. Unless they are the same as i, j and k, there's no reason why the matrix should be the identity matrix.

Secondly,

2) a)By considering the effects on the unit vectors (1,0,0),(0,1,0), (0,0,1) show what transformations are represented by the matrices.

A=[cos0,sin0,0;-sin0,cos0,0;0,0,1]

and

B=[cos0,0,sin0;0,1,0;-sin0,0,cos0]

(0 = theta)

am I right in thinking that B is a rotation in the y axis with angle theta? Yes

...

Is matrix A a rotation about the z axis with angle theta? Yes

Thus making A then B a rotation about the x axis? No. A rotation about the z-axis followed by a rotation about the y-axis does not give a rotation about the x-axis.