`star` Introduction

`star` Coordinate Axes and Coordinate Planes in Three Dimensional Space

`star` Coordinates of a Point in Space

`star` Coordinate Axes and Coordinate Planes in Three Dimensional Space

`star` Coordinates of a Point in Space

`\color{red} ✍️ ` You may recall that to locate the position of a point in a plane, we need two intersecting mutually perpendicular lines in the plane. These lines are called the `color(blue)"coordinate axes"` and the two numbers are called the `color(blue)"coordinates of the point with respect to the axes."`

`\color{red} ✍️ ` In actual life, we do not have to deal with points lying in a plane only. For example, consider the position of a ball thrown in space at different points of time or the position of an aeroplane as it flies from one place to another at different times during its flight.

`\color{red} ✍️ ` In actual life, we do not have to deal with points lying in a plane only. For example, consider the position of a ball thrown in space at different points of time or the position of an aeroplane as it flies from one place to another at different times during its flight.

`\color{red} ✍️ ` Consider three planes intersecting at a point `O` such that these three planes are mutually perpendicular to each other (Fig 12.1).

`\color{red} ✍️ ` These three planes intersect along the lines `X′OX, Y′OY` and `Z′OZ,` `color(blue)"called the x, y and z-axes"`, respectively.

`\color{red} ✍️ ` We may note that these lines are mutually perpendicular to each other. These lines constitute the rectangular coordinate system.

`\color{red} ✍️ ` The planes XOY, YOZ and ZOX, called, respectively the XY-plane, YZ-plane and the ZX-plane, are `color(blue)"known as the three coordinate planes."`

`\color{red} ✍️ ` The distances measured from `XY`-plane upwards in the direction of `OZ` are taken as positive and those measured downwards in the direction of `OZ′` are taken as negative.

`\color{red} ✍️ ` Similarly, the distance measured to the right of `ZX`-plane along `OY` are taken as positive, to the left of `ZX`-plane and along `OY′` as negative,

`\color{red} ✍️ ` In front of the `YZ`-plane along OX as positive and to the back of it along `OX′` as negative.

`\color{red} ✍️ ` The point `O` is called the `color(red)"origin of the coordinate system."`

`\color{red} ✍️ ` The three coordinate planes divide the space into eight parts `color(blue)("known as octants.")`

These octants could be named as `XOYZ,` `X′OYZ,`` X′OY′Z,`` XOY′Z, ``XOYZ′,`` X′OYZ′, ``X′OY′Z′` and `XOY′Z′`. and denoted by `I, II, III, ..., VIII , `respectively.

`\color{red} ✍️ ` These three planes intersect along the lines `X′OX, Y′OY` and `Z′OZ,` `color(blue)"called the x, y and z-axes"`, respectively.

`\color{red} ✍️ ` We may note that these lines are mutually perpendicular to each other. These lines constitute the rectangular coordinate system.

`\color{red} ✍️ ` The planes XOY, YOZ and ZOX, called, respectively the XY-plane, YZ-plane and the ZX-plane, are `color(blue)"known as the three coordinate planes."`

`\color{red} ✍️ ` The distances measured from `XY`-plane upwards in the direction of `OZ` are taken as positive and those measured downwards in the direction of `OZ′` are taken as negative.

`\color{red} ✍️ ` Similarly, the distance measured to the right of `ZX`-plane along `OY` are taken as positive, to the left of `ZX`-plane and along `OY′` as negative,

`\color{red} ✍️ ` In front of the `YZ`-plane along OX as positive and to the back of it along `OX′` as negative.

`\color{red} ✍️ ` The point `O` is called the `color(red)"origin of the coordinate system."`

`\color{red} ✍️ ` The three coordinate planes divide the space into eight parts `color(blue)("known as octants.")`

These octants could be named as `XOYZ,` `X′OYZ,`` X′OY′Z,`` XOY′Z, ``XOYZ′,`` X′OYZ′, ``X′OY′Z′` and `XOY′Z′`. and denoted by `I, II, III, ..., VIII , `respectively.

`\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)"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} ✍️ ` 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).`

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.

Q 3069534415

In Fig if P is (2,4,5), find the coordinates of F.

For the point F, the distance measured along OY is zero. Therefore, the coordinates of F are (2,0,5).

Q 3029634511

Find the octant in which the points (–3,1,2) and (–3,1,– 2) lie.

From the Table the point (–3,1, 2) lies in second octant and the point (–3, 1, – 2) lies in octant VI.