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‫Welcome back.

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‫So let's try to rewrite this system of equations in a different form.

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‫How about if we try to represent this relationship in a vector matrix form?

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‫If you do that, then it will look like this and you will have a matrix here that we will call R, and

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‫if you write it out, you will get the same expression like this one.

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‫And this Matrix R is called a rotation matrix in 2D, a two dimensional rotation matrix.

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‫It's called a rotation matrix because it's like rotating the body frame with respect to the inertia

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‫frame by an angle BPCI.

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‫However, what it really is, is just a matrix that transfers information from a body frame to the inertial

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‫frame.

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‫If you have a vector either position velocity or acceleration vector, which is represented in the body

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‫frame and you multiply a rotation matrix by it, then you get the same vector only in the inertial frame.

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‫Take a look at this position vector p.

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‫The vector itself does not change.

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‫What changes is how we represent it, either with inertial X and Y dimensions or body frame X and Y

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‫dimensions.

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‫And we can move back and forth between these two representations using the rotation matrix and its inverse

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‫like this.

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‫These are your P components in the inertia frame axis here and P Y is here and these are your P vector

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‫components in the body frame, your P body frame axis here, your P body frame y axis is here.

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‫And so I take this rotation matrix, which is this one, and I multiplied by this body frame position

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‫vector and I get the position vector in the inertia frame and if I want to do it in reverse then I need

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‫an inverse rotation matrix.

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‫And there you have it.

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‫I multiply the inverse of the rotation matrix R by the position vector in the inertia frame and I get

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‫the same position vector only represented in the body frame.

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‫I can abbreviate the inertia frame with the letter E because we can also call it the earth frame and

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‫I can abbreviate the body frame with the letter B and so if I put a superscript E here, then that means

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‫that the position vector is represented in the initial frame.

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‫And if I put superscript B here, then that means that the position vector is represented in the body

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‫frame.

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‫OK, that was a two dimensional case.

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‫Let's now expand this concept to 3D.

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‫If you look at one reference frame, rotate with respect to another in 2D like this.

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‫In other words, you have the body frame rotating with respect to the inertial frame, then what do

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‫you see when you now think in 3D?

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‫What axis are you rotating about?

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‫Give you this more thought and I will tell it to you in the next video.