My book states the following:

Two position vectors v and w are equal if and only if the terminal point of v is the same as the terminal point of w. This leads to the following result:

Theorem: Equality of Vectors

Two vectors v and w are equal if and only if their corresponding components are equal. That is,

If v = <a1, b1> and w = a2, b2>

then v = w if and only if a1 = a2 and b1 = b2

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The first part basically stated that only the terminal points of two vectors have to be the same for them to be equal. However, there can be two vectors with different initial points and the same terminal point, but this won't make the definition (second statement) true.

Why does the statement and the definition seem to contradict each other?

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Btw, the following was stated at the beginning of the section while describing geometric vectors:

"Two vectors v and w are equal, written v = w, if they have the same magnitude and direction. For example, the three vectors shown in figure 42 have the same magnitude and the same direction, so they are equal, even though they have different initial points and different terminal points. As a result, we find it useful to think of a vector simply as an arrow, keeping in mind that two arrows (vectors) are equal if they have the same direction and the same magnitude (length)."

This description does not match up with the ones described above, so I think it is safe to say that the material is bot referring to geometric vectors.

Lastly, keep in mind that the section only describes position vectors and geometric vectors.