# Expressing rotations in another, differently oriented coord system

#### Itsallmathintheend

##### New member
Hello math lovers,

I have a question about expressing x and y rotations one coord system, expressed in another coord system that is rotated about the z-axis:

A first right-handed coordinate system is rotated rx1 about its x1 axis and ry1 about its y1 axis.

A second right-handed coordinate system is rotated about the z1 axis by rz1 (relative to the first system).
The second system, in its turn, is rotated around its own axes, by rx2 and ry2.

What are rx2 and ry2 in the second system, expressed in rx1 and ry1 relative to the first system?

For example: a table can be tipped left/right and forwards/backwards. These tipping angles are known. Flat(!) on this table sits a toy car at certain, known angle relative to the table. At what angles is the toy car tipped/rotated expressed in angles relative to the table?

Thanks for your help and time.

A.

#### Subhotosh Khan

##### Super Moderator
Staff member
Hello math lovers,

I have a question about expressing x and y rotations one coord system, expressed in another coord system that is rotated about the z-axis:

A first right-handed coordinate system is rotated rx1 about its x1 axis and ry1 about its y1 axis.

A second right-handed coordinate system is rotated about the z1 axis by rz1 (relative to the first system).
The second system, in its turn, is rotated around its own axes, by rx2 and ry2.

What are rx2 and ry2 in the second system, expressed in rx1 and ry1 relative to the first system?

For example: a table can be tipped left/right and forwards/backwards. These tipping angles are known. Flat(!) on this table sits a toy car at certain, known angle relative to the table. At what angles is the toy car tipped/rotated expressed in angles relative to the table?

Thanks for your help and time.

A.
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#### Itsallmathintheend

##### New member

A large vessel carries a smaller vessel on her deck.

The heading of the smaller vessel is different to that of the main vessel; i.e. the smaller vessel points in a random direction on the deck of the larger vessel.

The large vessel rolls and pitches.

Given:
1) right handed coordinate systems
3) roll is a rotation about the y-axis and
4) pitch is a rotation about the x-axis.

Express the roll and pitch of the smaller vessel in:
1) the heading difference between large and small vessel, and
2) roll and pitch of the large vessel.

I have tried to solve this with unit vectors and right-handed 3d rotation matrices, working back from a point on the smaller vessel, given a smaller vessel's roll and pitch, and tried to express that outcome in roll and pitch of the larger vessel.

#### Subhotosh Khan

##### Super Moderator
Staff member

A large vessel carries a smaller vessel on her deck.

The heading of the smaller vessel is different to that of the main vessel; i.e. the smaller vessel points in a random direction on the deck of the larger vessel.

The large vessel rolls and pitches.

Given:
1) right handed coordinate systems