Engineering Mechanics Professor Manoj K Harbola Department of Physics Indian Institute of Technology Kanpur Module 1 Lecture No 05 Transformation of vectors under rotation (Refer Slide Time: 0:15) So so far we have just looked at the vector quantity as something that has a direction and that has a magnitude but now we are going to look at some more properties. (Refer Slide Time: 0:27
And one property that we specifically look at is the change of vector components under rotation. Let us see that. (Refer Slide Time: 0:33) Suppose you have a friend who asks you, how far is the house of another friend? And you say, it is 500 m or you say, you walk 300 m to the east I am just taking the previous example, and 400 m to the north. So that the direct distance of that house is 500 m. But you have to walk in a zigzag manner. In that you have to go east 1st and then north. Total distance you cover is 700 m. Now suppose you have another friend who is looking at it from a different frame. So for the previous friend, you had the X axis along the east and Y axis along the north. But another friend has his axis X prime and this at some angle Theta from the east and Y prime at some angle Theta from the north. Although the distance travelled by this person in going to the other house is going to be 700 m in both the frames, however this vector you can see is going to have components which are not going to be 300 and 400 m. But this is going to be its X component and this is going to be its Y component. So although the vector remains the same, it is 500 m in this north-east direction but its components along different frames are different. On the other hand, for the scalar quantity, distance, that remains the same, unchanged. So what we conclude? A scalar quantity which is nothing but a number remains unchanged in the two frames.
(Refer Slide Time: 3:06) On the other hand we also see that the components of a vector quantity change when we go from one frame, say XY to another frame which is rotated with respect to the 1st frame. X prime, Y prime which is rotated by an angle Theta with respect to the 1st frame. For the same vector quantity which in space is still pointing in the same direction, has the same magnitude. This is going to be X component, this is going to be Y component. On the other hand, the X prime component is going to be this and Y prime component is going to be this. And the relationship between X prime and Y prime component and X and Y component is well-known and that is what we are going to derive now. So a vector quantity must follow that relationship. Its components should change according to that relationship when we go from one frame to another frame which is rotated with respect to the 1st frame. So let us derive the relationship.
(Refer Slide Time: 4:21) So let us say there is a vector pointing in a certain direction in the original frame X and Y. So that this is its X component and this is its Y component. I am looking at the same vector from another frame. Let me now use a different colour. X prime and Y prime which is rotated with respect to the 1st frame by an angle Theta. The component X prime is going to be given here. This is going to be X prime and this is going to be Y prime. You can also see that this angle is also Theta. Let me also draw a perpendicular from here to here. This angle is also Theta. Let this point be A, B, C, this is the origin. Let this point be B prime, let this point be C prime. So we see that X prime component is going to be equal to OC prime + C prime B prime, that is this + this. And let me also call this point where it intersects as D. OC prime OC prime is nothing but X cosine theta. So this is going to be X cosine of theta. + C prime B prime is nothing but C prime D + DB prime. So this is nothing but C prime D + D B prime which is equal to X cosine of theta. C prime D is nothing but DB sine of theta. DB sine of theta. I am deliberately writing it like this. + DB prime AD sine of theta. Sine of theta I have taken out. Now you see, DB + AD is nothing but Y. So I can write this as X cosine of theta + Y sine of theta.