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Three Dimensional Geometry

Rectangular Coordinate System

Rectangular Coordinate System

If we draw three planes intersecting at O such that they are mutually perpendicular to each other, then these will intersect along the lines X′OX, Y′OY and Z′OZ. These lines constitute the rectangular coordinate system and are respectively known as the x, y and z-axes.

The point O is called the origin of the coordinate system.

Note:

The planes XOY, YOZ, and ZOX are known as the three coordinate planes and are respectively called the XY-plane, the YZ-plane and the ZX-plane.

The distances measured from XY-plane upwards in the direction of OZ are taken as positive and those measured downward in the direction of OZ′ are taken as negative.

The distances measured to the right of ZX-plane along OY are taken as positive and those measured to the left of ZX-plane along OY′ are taken as negative.

The distances measured in front of YZ-plane along OX are taken as positive and those measured at the back of YZ-plane along OX′ are taken as negative.

Coordinates of a Point in Space

If a point A lies in the first octant of a coordinate space, then the lengths of the perpendiculars drawn from point A to the planes XY, YZ and ZX are represented by x, y and z respectively and are called the coordinates of point A. This means that the coordinates of point A are (x, y, z). However, if point A would have been in any other quadrant, then the signs of x, y and z would change accordingly.

The coordinates of the origin are (0, 0, 0).

The coordinates of a point lying on different axes are as follows:

The coordinates of a point lying on the x-axis will be of the form (x, 0, 0).

The coordinates of a point lying on the y-axis will be of the form (0, y, 0).

The coordinates of a point lying on the z-axis will be of the form (0, 0, z).

The coordinates of a point lying on different planes are as follows:

The coordinates of a point lying in the XY-plane will be of the form (x, y, 0).

The coordinates of a point lying in the YZ-plane will be of the form (0, y, z).

The coordinates of a point lying in the ZX-plane will be of the form (x, 0, z).

The three coordinate planes divide the space into eight parts known as octants. These octants are named as XOYZ, X′OYZ, X′ OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′ and are denoted by I, II, III, IV, V, VI, VII and VIII respectively.

The sign of the coordinates of a point determines the octant in which the point lies. The following table shows the signs of the coordinates in the eight octants.

Octants →

I

II

III

IV

V

VI

VII

VIII

Coordinates↓

x

+

+

+

+

y

+

+

+

+

z

+

+

+

+

Distance Formula

The distance formula that is used for finding the distance between two points A(x1, y1, z1) and B(x2, y2, z2) lying in three-dimensional space is given by

AB =

Example: The distance between points A(3, 7, 8) and B(5, 0, 1) is given by.

Section Formula

The coordinates of point that divide the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) internally in the ratio m : n are given by .

If X is the mid-point of the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2), then X divides AB in the ratio 1 : 1. Hence, by using the section formula, the coordinates of point X will be given by .

The coordinates of point that divide the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) externally in the ratio m : n are given by.

If a point divides the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) internally in the ratio k : 1, then, by using the section formula, the coordinates of point will be given by

.

Centroid of a Triangle

The point of intersection of the medians of a triangle is called the centroid of the triangle. The coordinates of the centroid of a triangle whose vertices are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) are given by .

Area of a Triangle The area of a triangle whose vertices are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is given by 12x1y11x2y21x3y312+y1z11y2z21y3z312+z1x11z2x21z3x312. Reality checkQuestion:1

Use the following information to answer the next question.

A point with coordinates lies in the first octant. The point is rotated about the origin such that it now lies in VIII octant and its coordinates are (y − 2, 2y − 7x, y − 6x).

What are the coordinates of the point in the VI octant?

A)

(2, −5, 4)

B)

(−2, 3, −5)

C)

(−4, −7, 5)

D)

(−3, 4, −7)

A point with coordinates (a, b, c) in Ist octant has coordinates (a, - b, − c) in VIII octant.

The coordinates of given point in Ist octant are .

In the VIII octant, the coordinates should be .

However, it is given that in VIII octant, the coordinates are .

∴ 2x − 1 = y − 2

⇒ 2x − y = − 1 … (1)

− (2y − 3x) = 2y − 7x

⇒ − 5x + 2y = 0 … (2)

− (2y − 3) = y − 6x

⇒ 2x − y = − 1 … (3)

Solving equations (1) and (2), we obtain

x = 2 and y = 5

Thus, coordinates of the given point in the Ist octant are (2(2) −1, 2 (5) − 3 (2), 2 (5) − 3) = (3, 4, 7)

The point with coordinates (a, b, c) in Ist octant has coordinates (− a, b, − c) in VI octant.

Thus, in VI octant, the coordinates of the given point are (−3, 4, −7).

Question:2

If the two vertices of a triangle are (−2, 7, 3) and (1, −4, 1) and the centroid of the triangle is the origin, then what are the coordinates of the third vertex?

A)

B)

C)

(1, −3, −4)

D)

(2, −6, −8)

The two vertices of the triangle are given as (−2, 7, 3) and (1, −4, 1).

Let the coordinates of the third vertex be (x, y, z).

It is known that the coordinates of the centroid of the triangle, whose vertices are (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), are.

Therefore, the coordinates of the centroid of the given triangle are given by

Since the centroid of the given triangle is the origin (0, 0, 0), we obtain

Thus, the coordinates of the third vertex are (1, −3, −4).

Question:3

What is the area of the quadrilateral formed by the vertices (7, −7, 4), (1, −5, 1), (4, 1, 3), and (10, −1, 6), where vertices are taken in order?

A)

49unit2

B)

61unit2

C)

D)

Let PQRS be the quadrilateral formed by the vertices P (7, −7, 4), Q (1, −5, 1),

R (4, 1, 3), and S (10, −1, 6).

By distance formula, we obtain

It is observed that PQ = QR = RS = PS

Therefore, PQRS is a rhombus.

It can be observed that,

Thus, area of rhombus PQRS

Question:4

What is the area of the triangle formed by vertices (a, b, c), (b, c, a) and (c, a, b)?

A)

B)

C)

D)

Let ΔPQR be the triangle formed by vertices P (a, b, c), Q (b, c, a) and R (c, a, b).

The distance between points (x1, y1, z1) and (x2, y2, z2) is .

Hence, ΔPQR is equilateral.

∴Area of ΔPQR

Thus, the area of the given triangle is .