Vector and frame of reference?

My textbook has stated the following:

"If the frame of reference is translated or rotated, the vector doesn't change."

Although the length of the vector won't change, the angle that this vector makes with the positive direction of the newly defined x-axis changes, no? Hence, how is it possible to state that the vector doesn't change? Have I misunderstood the basic definition of a vector?

Please share your insights. Much thanks in advance :) Regards.

Answer

This came as a consequence of the basic idea that a vector obeying transformation laws is basically a tensor.

Translation and Rotation of inertial frame

One observes a vector based on his inertial frame. Even though different observers describe it in different ways, they all agree with the same value and direction.

When an inertial frame itself is translated, the vector defined in that frame also remain unchanged. For the simplest example take the position difference vector between two points, $\vec{r}=\vec{r_1}-\vec{r_2}$, where $\vec{r_1}$ and $\vec{r_2}$ are the position vectors from the origin of the frame. Even though the individual position vectors, $\vec{r_1}$ and $\vec{r_2}$ change during translation, since the origin gets shifted during translation, the position difference $\vec{r}=\vec{r_1}-\vec{r_2}$ remains unchanged.

This means, when a frame is translated, the vector remain unchanged, but it's components get transformed in such a way that the vector as a whole suffers no change. A vector is characterized by magnitude and direction. A simple translation does not affect the magnitude of the vector, but changes the values of the components without affecting the magnitude of the vector.

For example, the vector $\vec{r}=rcos\theta\hat{i}+rsin\theta\hat{j}$ has a magnitude of $r$ and lies in the xy plane characterized by the argument $\theta$ and the length of the vector $r$. If you shift the origin of the frame (or translate it), then the angle will contain a phase factor, say $\phi$, so that the vector modifies to $\vec{r'}=rcos(\theta +\phi)\hat{i}+rsin(\theta+\phi)\hat{j}$. Still the length of the vector is $r$. Also the phase factor does not affect it's direction. The angle changed, but the vector as a whole suffered no change.

Now, in the case of rotation about a fixed origin, obviously the components of the vector will get modified, because the angles change. But, the length of the vector is invariant under such a transformation. You imagine the second hand of a clock. Suppose it represents a vector. When it is rotated about its origin, the length is unchanged and the vector direction is too unchanged, because the change in orientation of the vector is contained in the change of components of the vector. The above example again helps here.

So, vectors (or physical quantities that are vectors) remain unchanged or they remain independent of the observers or frame of references. They have to be so, because physical quantities when related among each other forms a law and physical law are invariant no matter who observes it or in what way he observes it. This shows the symmetry of physical laws. This is the essence of relativity. But all vectors need not be like this. For example, the position vector do not obey vector transformation law, but is still a vector and not a tensor. But you know, in physics we are not interested in absolute quantities, but relative quantities. So physical quantities are of importance that obeys certain transformation laws assigned to them.