3D Geometry is used to represent a point, a line, or a plane with reference to the xaxis, yaxis, and zaxis respectively. The threedimensional geometry has all the concepts similar to the twodimensional coordinate geometry.
The 3d geometry helps in the representation of a line or a plane in a threedimensional plane, using the xaxis, yaxis, zaxis. The coordinates of any point in threedimensional geometry have three coordinates, (x, y, z).
The threedimensional cartesian coordinate system consists of three axes, the xaxis, the yaxis, and the zaxis, which are mutually perpendicular to each other and have the same units of length across all three axes. Similar to the twodimensional coordinate system, here also the point of intersection of these three axes is the origin O, and these axes divide the space into eight octants. Any point in 3D Geometry is represented with the coordinates (x, y, z).
Further the coordinates of a points in the eight octants are (+x,+y,+z), (x,+y,+z), (+x,+y,z), (x,+y,z), (+x,y,+z), (x,y,+z), (+x,y,z), (x,y,z).
Notation of a point in a cartesian coordinate system is a way of presenting a point for easy understanding and calculations. The points in a cartesian coordinate system are written in parentheses, and separated by a comma. The examples of a point in a threedimensional frame is (2, 5, 4). The origin is denoted by the O and the coordinates of a point is denoted by the point (x, y, z). Here the last alphabets of the alphabetical series are taken or the first alphabets of the word is taken to represent the coordinates of a point.
A coordinate is an address, which helps to locate a point in space. For a threedimensional frame, the coordinates of a point is (x, y, z). Here let us take note of these three important terms.
The 3D geometry makes use of the three coordinates to represent a point. The important concepts with reference to threedimensional geometry are direction ratio, direction cosine, distance formula, midpoint formula, and section formula. The following are the important concepts of 3D Geometry.
Direction Ratios
The point A,(a, b, c) is represented as a vector with the position vector as → O A = a → i + b → j + c → k and has the direction ratios a, b, c. This ratio represents the vector line with reference to the xaxis, yaxis, and zaxis respectively. Further, these direction ratios also help to derive the direction cosines.
Direction Cosine
Direction Cosine gives the relation of a vector or a line in a threedimensional space, with each of the three axes. The direction cosine is the cosine of the angle subtended by this line with the xaxis, yaxis, and zaxis respectively. If the angles subtended by the line with the three axes are α, β, and γ, then the direction cosines are Cosα, Cosβ, Cosγ respectively. is The direction cosines are also represented by l, m, n, and we can prove that l^{2} + m^{2} + n^{2} = 1.
Distance Formula
The distance between two points (x_{1} , y_{1} , z_{1}) and x_{2}, y_{2}, z_{2}) is the shortest distance, and is equal to the square root of the summation of the square of the difference of the x coordinates, the ycoordinates, and the zcoordinates of the two given points. The formula for the distance between two points is as follows.
MidPoint Formula
The formula to find the midpoint of the line joining the points (x_{1}, y_{1}, z_{1}) and x_{2}, y_{2}, z_{2}) is a new point, whose abscissa is the average of the x values of the two given points, and the ordinate is the average of the y values of the two given points. The midpoint lies on the line joining the two points and is located exactly between the two points.
Section Formula
The section formula is useful to find the coordinates of a point that divides the line segment joining the points ( x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) in the ratio m : n . The point dividing the given two points lies on the line joining the two points and is available either between the two points or on the line, beyond the two points.
The three dimensional geometry is used for the representation of a point, line, or a plane. Let us check the different forms of representation of a point, line, and plane in threedimensional geometry.
Representation of a point in 3D Geometry
The point in a threedimensional geometry can be represented either in cartesian form or a vector form. The two forms of representation of the point in a 3D geometry are as follows.
Cartesian Form: The cartesian form of representation of any point in 3D geometry uses three coordinates with reference to the xaxis, yaxis, and zaxis respectively. The coordinates of any point in a 3D geometry is (x, y, z). The x value of the point is called the abscissa, the y value is called the ordinate, and the z value is called the applicate.
Vector Form: The vector form of representation of a point P is a position vector OP, and is written as → O P = x → i + y → j + z → k , where → i , → j , → k are the unit vectors along the xaxis, yaxis, and zaxis respectively.
The equation of a line in a threedimensional cartesian system can be computed from the following two methods. The two methods of finding the equation of a line are as follows.
The equation of a plane in a cartesian coordinate system can be computed through different methods based on the available inputs values about the plane. The following are the four different expressions for the equation of a plane.
406 videos217 docs164 tests


Explore Courses for Class 10 exam
