Mathematics > Three Dimensional Geometry

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  • 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.
    • Any point on x-axis is of the form (x, 0, 0)
    • Any point on y-axis is of the form (0, y, 0)
    • Any point on z-axis is of the form (0, 0, z).
  • Distance between two points P(x1, y1, z1) and Q (x2, y2, z2) is given by

    PQ =

  • The coordinates of the point R which divides the line segment joining two points  and Q () internally and externally in the ratio m : n are given by

      and

  • The coordinates of the mid-point of the line segment joining two points P() and Q() are

    [(x1+x2)/2, (y1+y2)/2, (z1+z2)/2]

  • The coordinates of the centroid of the triangle, whose vertices are () () and (), are

    [, , ]

 

Examples

Question

Find the distance between the points P(1, –3, 4) and Q (– 4, 1, 2).

Solution

The distance PQ between the points P (1,–3, 4) and Q (– 4, 1, 2) is

PQ = (-4-1)2 + (1-(-3))2 + (2-4)2

= 45 units

 

Question

Find the equation of the set of the points P such that its distances from the points A (3, 4, –5) and B (– 2, 1, 4) are equal.

Solution

 If P (x, y, z) be any point such that PA = PB

(x − 3)2 + (y − 4)2 + (z + 5)2(x + 2)2 + (y −1)2 + (z − 4)2

(x − 3)2 + ( y − 4)2 + (z + 5)2 =  (x + 2)2 + (y −1)2 + (z − 4)2

10    x + 6y – 18z – 29 = 0

 

Question

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, – 6), respectively, find the coordinates of the point C.

Solution

Let the coordinates of C be (x, y, z) and the coordinates of the centroid G be (1, 1, 1). Then

 = 1

x = 1

 = 1

y = 1

 = 1

z = 2

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