‫Welcome back.
‫So let's talk about the cross product applications now, one of the applications is actually a general
‫moment calculation formula.
‫So far we have computed moments in 2D, right.
‫Let's imagine a beam like in the assignment in the first part of the core series.
‫In that assignment, the beam initially hung on to courts like this.
‫This is the beam here and these are the two courts and this is the ceiling f g is the force of gravity
‫here that acts through the center of mass of the beam, which is here.
‫Then the left cord breaks and the beam starts falling down.
‫And you have to compute the moment about the point B right here.
‫When you did that, then everything happened on an X Y plane.
‫You had Y here and you had X here.
‫That is because we viewed our world in 2D.
‫However, in 3D, everything is much more general.
‫In 2D, the direction of the moment was either popping out of the plane towards you like this or away
‫from you like this, according to the right hand rule.
‫That is because that is how we measure the direction of rotations in the first place.
‫The rotation vector, whether it's angular velocity, angular acceleration or movement, it's vector
‫is perpendicular to the plane on which the rotation is happening.
‫So if something rotates like this, like you see here, then the vector of rotation is the red one here.
‫That's the direction of the rotation, the red vector.
‫Since moments cause rotation there, vectors are represented in the same way.
‫Now, in 3D, your moment vector can have infinite directions in a 3D space, you can imagine a sphere
‫here and the moment the vector can point in any direction on the sphere.
‫So instead of just saying moment equals distance times force like we did in the previous course in this
‫core series, we now have to go more generic in a 3D space.
‫We define a 3D position vector like this.
‫Our vector equals our X or Y and our Z transposed in a 3D force vector is like this affects F, Y and
‫Z transposed.
‫And of course this would be in meters and this would be in Newton's and the moment in 3D about a certain
‫point.
‫For example, Point O is calculated like this, the moment vector about the point O equals the position
‫vector cross the force vector.
‫That's the generic formula of moment calculation where you need the cross product and the cross product
‫here makes sure that only the component of the force vector that is perpendicular to the position vector
‫is taken into account.
‫Remember, that is what the cross product did.
‫It only multiplied the components of two vectors that were perpendicular to each other in the direction
‫of the moment.
‫Vector is perpendicular to both of them.
‫It's perpendicular to the position vector and it's also perpendicular to the force vector in 3D.
‫It looks something like this.
‫This is the point o here.
‫This is the position vector here.
‫The position vector could be the distance from the center of the drone to the propeller, for example.
‫So this is the center of the drone here and this is where the propeller is.
‫And then you also have the force vector here and this force vector could be the thrust of the propeller.
‫And so the drone experience is a moment m about its center that makes it angular accelerate about point.
‫Oh, like this.
‫So then of course means that this reference frame in white is the inertial reference frame and then
‫the body frame that is attached to the drone is this one, for example, small X that goes along this
‫position vector.
‫Be careful here though.
‫Order matters a krosby is not be cross a in fact A cross B equals minus B cross A and you saw it before
‫with the right hand rule as well.
‫To compute the moment you have to compute moment vector equals the position vector, cross the force
‫vector and not the other way around.
‫Force vector cross the position vector would be wrong, it would give you a wrong direction.
‫In fact it would give you a direction that is opposite to the right direction.
‫And again, remember, this green movement vector is perpendicular to both the position vector and to
‫the force vector.
‫If we shift this force vector here just for comparison.
‫So this green movement vector, as you can see, it forms a ninety degree angle with the position vector
‫and with the force vector.
‫And so the moment vector can also be written like this.
‫You have X and Y and then M, Z and then you transpose it.
‫So in the end, these are all three dimensional vectors here.
‫And so in a 2D case, it didn't matter whether you wrote F times the or the Times F that is because
‫we multiply two kilos with each other.
‫But in 3D, if you mess up the order, then your moment will be in the opposite direction because F
‫Cross R would give you minus M about the point o this would be the right.
‫Answer here.
‫So that has been one application of across product in the next video, we will talk about another one.
‫See you there.