‫Welcome back.
‫So let's try to rewrite this system of equations in a different form.
‫How about if we try to represent this relationship in a vector matrix form?
‫If you do that, then it will look like this and you will have a matrix here that we will call R, and
‫if you write it out, you will get the same expression like this one.
‫And this Matrix R is called a rotation matrix in 2D, a two dimensional rotation matrix.
‫It's called a rotation matrix because it's like rotating the body frame with respect to the inertia
‫frame by an angle BPCI.
‫However, what it really is, is just a matrix that transfers information from a body frame to the inertial
‫frame.
‫If you have a vector either position velocity or acceleration vector, which is represented in the body
‫frame and you multiply a rotation matrix by it, then you get the same vector only in the inertial frame.
‫Take a look at this position vector p.
‫The vector itself does not change.
‫What changes is how we represent it, either with inertial X and Y dimensions or body frame X and Y
‫dimensions.
‫And we can move back and forth between these two representations using the rotation matrix and its inverse
‫like this.
‫These are your P components in the inertia frame axis here and P Y is here and these are your P vector
‫components in the body frame, your P body frame axis here, your P body frame y axis is here.
‫And so I take this rotation matrix, which is this one, and I multiplied by this body frame position
‫vector and I get the position vector in the inertia frame and if I want to do it in reverse then I need
‫an inverse rotation matrix.
‫And there you have it.
‫I multiply the inverse of the rotation matrix R by the position vector in the inertia frame and I get
‫the same position vector only represented in the body frame.
‫I can abbreviate the inertia frame with the letter E because we can also call it the earth frame and
‫I can abbreviate the body frame with the letter B and so if I put a superscript E here, then that means
‫that the position vector is represented in the initial frame.
‫And if I put superscript B here, then that means that the position vector is represented in the body
‫frame.
‫OK, that was a two dimensional case.
‫Let's now expand this concept to 3D.
‫If you look at one reference frame, rotate with respect to another in 2D like this.
‫In other words, you have the body frame rotating with respect to the inertial frame, then what do
‫you see when you now think in 3D?
‫What axis are you rotating about?
‫Give you this more thought and I will tell it to you in the next video.