`\color{red} ✍️ ` Given a point `P` in space, we drop a perpendicular `PM` on the `XY`-plane with `M` as the foot of this perpendicular (Fig 12.2).
`\color{red} ✍️ ` Then, from the point `M`, we draw a perpendicular `ML` to the `x-`axis, meeting it at `L`.
`\color{red} ✍️ ` Let `OL` be `x`, `LM` be `y` and `MP` be `z.` Then `x`,`y` and `z` are called the `x`, `y` and `z` `color(blue)("coordinates")`, respectively, the octant `XOYZ` and so all `x, y, z` are positive.
`\color{red} ✍️ ` If `P` was in any other octant, the signs of `x, y` and `z` would change accordingly.
`\color{red} ✍️ ` Thus, to each point `P` in the space there corresponds an ordered triplet `(x, y, z)` of real numbers.
`color(red)("Alternatively")`, through the point `P` in the space, we draw three planes parallel to the coordinate planes, meeting the `x`-axis, `y`-axis and `z`-axis in the points `A, B` and `C,` respectively (Fig 12.3).
Let `OA = x, OB = y` and `OC = z.` Then, the point `P` will have the coordinates `x, y` and `z` and we write `P (x, y, z).`
`color(red)("Conversely")`, given `x, y` and `z`, we locate the three points `A, B` and `C` on the three coordinate axes.
Through the points `A, B` and `C` we draw planes parallel to the `YZ`-plane, `ZX`-plane and `XY`-plane, respectively.
The point of interesection of these three planes, namely, `ADPF, BDPE` and `CEPF` is obviously the point `P, ` corresponding to the ordered triplet `(x, y, z).`
We observe that if `P (x, y, z)` is any point in the space, then `x, y` and `z` are perpendicular distances from `YZ, ZX` and `XY` planes, respectively.
`color(red)"Note : "` The coordinates of the origin `O` are `(0,0,0).`
The coordinates of any point on the `x`-axis will be as `(x,0,0)` and the coordinates of any point in the `YZ`-plane will be as `(0, y, z).`
`color(red)"Remark "` The sign of the coordinates of a point determine the octant in which the point lies. The following table shows the signs of the coordinates in eight octants.
`\color{red} ✍️ ` Given a point `P` in space, we drop a perpendicular `PM` on the `XY`-plane with `M` as the foot of this perpendicular (Fig 12.2).
`\color{red} ✍️ ` Then, from the point `M`, we draw a perpendicular `ML` to the `x-`axis, meeting it at `L`.
`\color{red} ✍️ ` Let `OL` be `x`, `LM` be `y` and `MP` be `z.` Then `x`,`y` and `z` are called the `x`, `y` and `z` `color(blue)("coordinates")`, respectively, the octant `XOYZ` and so all `x, y, z` are positive.
`\color{red} ✍️ ` If `P` was in any other octant, the signs of `x, y` and `z` would change accordingly.
`\color{red} ✍️ ` Thus, to each point `P` in the space there corresponds an ordered triplet `(x, y, z)` of real numbers.
`color(red)("Alternatively")`, through the point `P` in the space, we draw three planes parallel to the coordinate planes, meeting the `x`-axis, `y`-axis and `z`-axis in the points `A, B` and `C,` respectively (Fig 12.3).
Let `OA = x, OB = y` and `OC = z.` Then, the point `P` will have the coordinates `x, y` and `z` and we write `P (x, y, z).`
`color(red)("Conversely")`, given `x, y` and `z`, we locate the three points `A, B` and `C` on the three coordinate axes.
Through the points `A, B` and `C` we draw planes parallel to the `YZ`-plane, `ZX`-plane and `XY`-plane, respectively.
The point of interesection of these three planes, namely, `ADPF, BDPE` and `CEPF` is obviously the point `P, ` corresponding to the ordered triplet `(x, y, z).`
We observe that if `P (x, y, z)` is any point in the space, then `x, y` and `z` are perpendicular distances from `YZ, ZX` and `XY` planes, respectively.
`color(red)"Note : "` The coordinates of the origin `O` are `(0,0,0).`
The coordinates of any point on the `x`-axis will be as `(x,0,0)` and the coordinates of any point in the `YZ`-plane will be as `(0, y, z).`
`color(red)"Remark "` The sign of the coordinates of a point determine the octant in which the point lies. The following table shows the signs of the coordinates in eight octants.