Help with vector basics

[heady haven]

Is this example in a mechanics textbook of vector coordinates wrong? I can’t tell if it is or if I’m missing something

[wicked moat]

$y'_o < y'_1$

[glad tree]

UnexpectedTaco

[wicked moat]

As you get closer to the $x'$ axis, magnitude of $y'$ decreases.

[glad tree]

UnexpectedTaco

[wicked moat]

You may confuse this with the $-x$ and $-y$ axis, but the point of this example is to illustrate how different ways of orientation of the axis can change the way the components of the vectors are determined.

[glad tree]

UnexpectedTaco

[wicked moat]

For a more immediate example consider the same xy plane, simply rotated by some angle. The way you'd write the rectangular components of the vector here is different from if the axis were the standard xy.

[heady haven]

I think I understand. The second coordinate system is defined such that the value of the y’ axis decreases towards the right

So I should be looking at how the vector is defined in (x’, y’) terms first to determine the orientation of the second space