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If $P(x_1, y_1)$ and $Q(x_2, y_2)$ are the two points on a plane, then the change in $x$ and $y$ coordinates is denoted by $∆x$ and $∆y$ respectively. Therefore, $x = ∆x = x_2 - x_1$ and $y = ∆y = y_2 - y_1$. The quantities $∆x$ and $∆y$ may be positive, negative or zero. For example, when $x_2 > x_1$ then $∆x$ is positive and if $x_2 < x_1$ then $∆x$ is negative.

I want to ask that if $x_1$ and $x_2$ both lie in the first quadrant then they will both be positive, and the change in $x$ will be positive. If we say that $x_1 = 4$ and $x_2 = 3$ then the change in $x$ will be negative. I think, the passage which I've quoted above says that if the point $Q$ lies in the $II$ or $III$ quadrant then $x_2 < x_1$, but if they $P$ and $Q$ lie in the $I$ quadrant then the value which is greater, say $Q$ will have $x_2$ coordinate, and the smaller one will have $x_1$ coordinate. Am I right? Please help me with my confusion.

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up vote 2 down vote accepted

Given: starting point: $(x_1, y_1) \longrightarrow$ ending point $(x_2, y_2)$:

$$\Delta x = x\text{-coordinate of ending point} - x\text{-coordinate of starting point}:\quad x_2 - x_1$$

$$\Delta y = y\text{-coordinate of ending point} - y\text{-coordinate of starting point}:\quad y_2 - y_1$$

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The points $P$ and $Q$ can lie anywhere. When we talk about the difference, we have a start point and an end point. If we start with $P(x_1,y_1)$ and end with $Q(x_2,y_2)$, which is what is assumed in the text, then the difference in the $x$-coordinate is $\Delta x=x_2-x_1$ and that in the $y$-coordinate is $\Delta y=y_2-y_1$. The vector $(\Delta x,\Delta y)$ then represents the change as one moves from $P$ to $Q$.

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and what if we start with $Q(x_2,y_2)$? – Samama Fahim Mar 29 '13 at 23:50
OK, I think $x_1$ is always supposed to be a start point and $x_2$ is always supposed to be an end point, and therefore $P$ and $Q$ might lie anywhere. Am I right? – Samama Fahim Mar 30 '13 at 0:02

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