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Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry : It was not until the middle of the nineteenth century that geometry was extended to more than three dimensions, the well-known application of which is in the Space-Time Continuum of Einstein’s Theory of Relativity.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry – syllabus

  • Introduction to Three-dimensional Geometry
  • Coordinate axes and coordinate planes in three dimensions.
  • Coordinates of a point.
  • Distance between two points and section formula and problems.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

In three-dimensional space, we identify a point with respect to the distances from the three reference planes.

Take two planes perpendicular to each other.

The intersection of these two perpendicular planes is a straight line.

Let’s take another plane perpendicular to both the existing planes.

These lines are called axes, and are named as X, Y and Z.

The point of their concurrence is called the origin.

The origin is denoted by the letter O. By convention, we consider the X—Y plane as the horizontal plane. Distances are measured starting from the origin.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Distances measured in one direction are taken as positive, and in the opposite side as negative.

For example, the distance measured from the origin along the X-axis along the direction shown is taken as positive.

The distance measured from the origin in the opposite side is considered negative.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

To indicate this, we assign the symbol X’.

A similar convention is followed for calculating the distances along the Y and Z axes.

The plane formed by the X and Y axes is called the XY plane.

Similarly, the plane formed by the Y and Z axes is called the YZ plane, and the plane formed by the Z and X-axes is called the ZX plane.

The XY plane, YZ plane and ZX plane divide space into eight octants.

XOYZ is an octant. This octant is enclosed by the XYZ axes.

Similarly, the other octants are named according to the axes that enclose them. Any point in space is represented with the help of the distances that it measures with respective to the planes. These distances are called the coordinates of the point.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Suppose we have a point (x, y, z) in space. The coordinates of this point are x, y, z.
Each coordinate individually indicates the shortest distance of the point from the corresponding plane.

The coordinate, x, determines the shortest distance of the point from the Y—Z plane.

Similarly, the y co-ordinate represents the distance of the point from the Z—X plane, and z represents the distance of the point from the X—Y plane.

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

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Using the coordinate axes in locating a point in the coordinate space:

First method is to drop a perpendicular on the X—Y plane. Let this perpendicular meet the X—Y plane at point Q.

Then we drop a perpendicular from Q on the X-axis. This meets at a point, say, R.
Distance OR is the x coordinate, RQ is the y coordinate and PQ is the z coordinate.

A point can be traced if the coordinates of a point are given.

First, identify a point, say R, on the X-axis, which is at distance x from the origin.
Next, we draw a perpendicular of length y from point R and parallel to the Y-axis on the X—Y plane to a point, say Q.

Lastly, we draw a perpendicular of length z from point R perpendicular to the X—Y plane.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Property among the coordinates of the points:

Any point that lies on the X-axis will have its coordinates in the form (x,0,0).
For example, a point at a distance of six units from the origin, on the x axis will have the coordinates (6,0,0)

Similarly, the points that lie on Y-axis will have coordinates of the form (0,y,0), and the points on the Z-axis will have coordinates of the form (0,0,z) Here, the letters, x, y and z belong to the set of real numbers.

Any point on the XY plane will be of the form,
 (x,y,0). The points (2, 2, 0), (6, -3, 0) lie on the XY plane.

Any point on the YZ plane will be of the form,
 (0,y,z) and a point on the ZX plane will be of the form (x,0,z). Here, the letters, x, y and z belong to the set of real numbers.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Locating a point using the intersection of planes:

First, take a plane at a distance x and parallel to the Y—Z plane.
Next, a plane parallel to the X—Z plane and at a distance of y is taken.

Finally, another plane parallel to the X—Y plane and at a distance of z is taken.

First, take two planes – one parallel to the Y—Z plane and the other parallel to the X—Z plane.

The intersection will be a line parallel the X-axis.

Likewise, the intersection of the other planes creates lines as shown.

The concurrence of these lines is point P.

The coordinate axes divide the coordinate space into eight parts, called octants. The sign of the coordinates determine the position of a point in a particular octant. For example, the point (2,-3, 3) lies in the fourth octant.

The x coordinate of the point is 2. So we measure 2 units along the X-axis and mark a point.

The y coordinate is -3. So, from this point, we measure (2, 3) on the X—Y plane in the direction of OY’ parallel to the Y-axis. We mark a point there.

For the z coordinate, that is, 3, we measure 3 units from the point perpendicular to the X—Y plane.

We can notice that the point lies in the fourth octant.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

The table shown gives the details of the octant according to the signs of the coordinates of a given point.

                \ Co-ordinates
Octants\
       I       II     III     IV      V       VI     VII     VIII
                x   +    –    –   +   +    –    –   +
                y   +   +    –    –   +   +    –    –
                z   +   +   +   +    –    –    –    –
 
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 coordinate axes and the two numbers are called the coordinates of the point with respect to the axes. 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. Similarly, if we were to locate the position of the lowest tip of an electric bulb hanging from the ceiling of a room or the position of the central tip of the ceiling fan in a room, we will not only require the perpendicular distances of the point to be located from two perpendicular walls of the room but also the height of the point from the floor of the room.
Therefore, we need not only two but three numbers representing the perpendicular distances of the point from three mutually perpendicular planes, namely the floor of the room and two adjacent walls of the room. The three numbers representing the three distances are called the coordinates of the point with reference to the three coordinate planes. So, a point in space has three coordinates.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

In this Chapter, we shall study the basic concepts of geometry in three dimensional space. In three dimensions, the coordinate axes of a rectangular Cartesian coordinate system are three mutually perpendicular lines. The axes are called the x, y and z-axes.

The three planes determined by the pair of axes are the coordinate planes, called XY, YZ and ZX-planes.
The three coordinate planes divide the space into eight parts known as octants.
The coordinates of a point P in three dimensional geometry is always written in the form of triplet like (x, y, z). Here x, y and z are the distances from the YZ, ZX and XY-planes.
(i) Any point on x-axis is of the form (x, 0, 0)
(ii) Any point on y-axis is of the form (0, y, 0)
(iii) Any point on z-axis is of the form (0, 0, z).

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

Distance between two points P(x1 , y1 , z1 ) and Q (x2 , y2 , z2 ) is given by PQ = √[(x2-x1)2+(y2-y1)2+(z2-z1)2]

The coordinates of the point R which divides the line segment joining two points P (x1 y1 z1 ) and Q (x2 , y2 , z2 ) internally and externally in the ratio m : n are given by {[(mx2+nx1)/(m+n)],[(my2+ny1)/(m+n)],[(mz2+nz1)/(m+n)]} and {[(mx2 – nx1)/(m-n)],[(my2 – ny1)/(m-n)],[(mz– nz1)/(m-n)]} 
respectively.

Karnataka Class 11 Commerce Maths Introduction to Three-dimensional Geometry

The coordinates of the mid-point of the line segment joining two points.

P(x1 , y1 , z1 ) and Q (x2 , y2 , z2 ) are [(x1+x2)/2, (y1+y2)/2,(z1+z2)/2]

The coordinates of the centroid of the triangle, whose vertices are (x1 , y1 , z1 ),(x2 , y2 , z2 ) and (x3 , y3 , z3 ) are [(x1+x2+x3)/3 , (y1+y2+y3)/3,(z1+z2+z3)/3]

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