‫Welcome back, I hope you tried the exercise yourself before watching this video, so we rotated pie
‫over to Radiance about the inertial Z axis and this is what we got again, p, q, r, r, one radians
‫per second.
‫So this would be P here, this would be Q and this would be R, so the resultant vector from the origin
‫will be something like this.
‫Now by inspection, you know that the coordinates of this vector in the inertial frame are one one and
‫then minus one.
‫With respect to the inertia frame, you have a component in the positive X direction, positive Y direction
‫and then negative Z direction in the body frame.
‫It would still be one, one and one.
‫But let's use our rotation matrix approach.
‫In the last video I showed you this matrix r z.
‫Since we rotated power to Radians about the inertial Z axis, that means that our BPCI equals PI over
‫to radians.
‫So if you substitute BPCI with PI over two, then you will get this matrix here to calculate our coordinates
‫in the inertia frame we write X, Y, Z, which are our coordinates in the inertia frame equals R,
‫Z times are Y times R, X, and then we multiply this product of matrices by the coordinates in the
‫body frame, in this body frame here in red.
‫From the last video, you know what this product is.
‫It's this one here.
‫And so we take our R Z matrix.
‫We multiply it by this matrix, which is our R Y times are X times the coordinate values in the body
‫frame.
‫And so if you multiply them all together, then you will get our Z times are Y times R X, which is
‫this one here.
‫And then you multiply that by this vector and you will end up with one one and minus one.
‫And as you can see, X equals one, Y equals one, and then Z equals minus one.
‫These are the inertia frame coordinates and you can also see it here by inspection.
‫And that's the purpose of rotation matrices.
‫Let's say that, you know, the coordinates of a vector in the body frame, which is our case here.
‫However, you would like to know the coordinates of the same vector, but in the initial frame, this
‫vector, the purple one, it's not changing.
‫It is what it is.
‫But if you first measured that vector in the body frame, then thanks to rotation matrices, you can
‫instantly compute the coordinates of the same vector, but in a different frame in the inertia frame.
‫And of course, if you're Phi Theta and PSI, if they're all pi over to radians, then you don't really
‫need rotation matrices.
‫But what if your, let's say PHI is five or six radians, then Theta is PI over three radians and then
‫BPCI is minus Pi over four radians.
‫Then you cannot visually inspected.
‫Right, but you can put your angles into the rotation matrices and then find the coordinates of the
‫vector in the inertia frame like this x, y, z equals then our Z where psi equals minus pi over four
‫radians times are Y where theta equals pi over three irradiance times are X, where phi equals five
‫or six radians times your angular velocity vector in the body frame you put your angles in, you will
‫have a three by three matrix here, which will be a product of.
‫These three matrices and then the vector that you had initially measured in the body frame, once you
‫multiply this product of matrices by the body frame coordinates, you will know the coordinates of that
‫vector in the initial frame.
‫So that's why they are used in engineering and that's why we will be using them in this course as well.
‫Thank you very much.