3. Transformation of vector components

Let us now see how the column form of a vector transform when we change the coordinate system just like we found the transformation for a second order tensor’s matrix representation. If we have a vector v, we can represent it in column form with respect to (e1, e2, e3) coordinate system as well as with respect to (ŕ1, ŕ2, ŕ3) coordinate system. To relate these two representations, one can easily deduce: (12)

When we change coordinates, the way we express vectors also changes. Regardless of the system we use (e1, e2, e3 or ŕ1, ŕ2, ŕ3), the underlying vector (v) remains the same. This transformation is reflected in how we write the vector in column form. The equation referenced in (12) shows the relationship between these two representations.

Imagine you are using different maps for navigation. The point you want to reach (the vector) stays constant, but the way you describe it (coordinates) changes based on which map you are looking at. Just like you can read a point's location using different map styles, we can express our vector in different coordinate systems while its essence remains unchanged.

What is to be noted here is that for transforming the vector components, we need to pre-multiply by [R]T while for relating the basis vectors, we need to pre-multiply by R (see equation (4)). This is because the vector v has to remain the same, we are just trying to write it in two different coordinate systems.

When transforming components of a vector, we use the transpose of the rotation matrix ([R]T). This ensures that the numerical values of the vector still represent the same physical vector. On the other hand, for transforming the basis vectors themselves, we directly use the rotation matrix (R). This reflects how the basis direction changes between coordinate systems without altering the vector itself.

Think of rotating a photograph. The picture (your vector) remains the same but how it sits on the screen (the coordinate system) can change depending on how you manipulate it (using R or [R]T). Just like you rotate the image but still see the same subject, we can adjust our vector's representation while keeping its true form unchanged.

Therefore, the basis vector ŕi gets transformed with rotation R whereas the components ŕi get transformed with RT. RTR gets cancelled and becomes I.

This section indicates that when we apply both the rotation of the basis vectors and the transformation of the vector components, a multiplication occurs between R and its transpose (RT). The neat aspect is that this combination of operations effectively cancels out, leading to the identity matrix (I), which represents no change in the fundamental properties of the vector.

It's like juggling two balls simultaneously. If you throw one ball up (R) and catch it back down (RT), you have effectively returned the system to its original state (I) because nothing has changed overall. This aspect of cancellation assures us that our vector hasn't altered, despite the transformations in representation.

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