Co-ordinate Geometry
279. The Order Matters · Why (3, -2) is not the same as (-2, 3)
Q is always (b, a) when P is (a, b).
In an ordered pair (x, y) the order matters: the first number is the x-coordinate (horizontal) and the second is the y-coordinate (vertical). So (a, b) and (b, a) are usually different points.
What this lesson covers
Try to break it
Drag P anywhere on the grid. P sits at (a, b); Q automatically lands at (b, a). The two points coincide only when a = b (on the line y = x). Drag P across that line and watch P and Q swap sides — they're always mirror images in the line y = x.
How you build it
Plot a point P and its coordinate-swap Q.
- Plot point P = (5, 2): from O, count 5 units right along the x-axis, then 2 units up, and click that grid point. Dashed lines to both axes appear so you can read off (5, 2).
- Now swap the order: Q = (2, 5). Count 2 right, then 5 up, and click. Its dashed guide lines appear too. Q lands in a different spot from P — so order matters: P ≠ Q.
The proof, step by step
Prove that the point (a, b) differs from the point (b, a).
- P is defined as (a, b) and Q is defined as (b, a).
- For P and Q to be the same point, their x-coordinates must match: a = b.
- Their y-coordinates must also match: b = a.
- Therefore, P and Q coincide if and only if a = b.
Worked example
In the Cartesian plane, point M is located at (p, q) and point N is located at (q, p). If M and N represent the exact same location, which of the following must be true?
Since M(p, q) and N(q, p) are the same point, their corresponding coordinates must be equal. Thus, p = q and q = p. This means the first and second components must be identical.
- p = -q
- p = q — correct
- p + q = 0
- p² = q