Inside Story

- 1 THREE DIMENSIONAL GEOMETRY
- 1.1 ORIGIN
- 1.2 COORDINATE AXES
- 1.3 COORDINATE PLANES
- 1.4 COORDINATES
- 1.5 DISTANCE FORMULA
- 1.6 VECTOR FORM
- 1.7 DIRECTION COSINES
- 1.8 STRAIGHT LINES
- 1.9 SKEW LINES

# THREE DIMENSIONAL GEOMETRY

## ORIGIN

## COORDINATE AXES

## COORDINATE PLANES

## COORDINATES

Let’s take a any point P in the space. Draw PL , PM and PN perpendicularly to the XY, YZ and ZX planes, then

### SIGN CONVENTION OF A POINT

## POINTS TO BE REMEMBERED |

1). The coordinates of any point on the X axes are of the form of (a,0,0) , similarly for the Y and Z axes is (0,b,0) and (0,0,c). 2). Coordinates of any point in the XY plane is given as (a,b,0) , similarly for YZ and ZX planes are (0,b,c) and (a,0,c). 3). The equation of the XY plane is given as z = 0, similarly for YZ plane is x = 0 and for ZX plane is y = 0. |

## DISTANCE FORMULA

#### DISTANCE FORMULA

#### DISTANCE FROM THE ORIGIN

## POINTS TO BE REMEMBERED |

1). The position vector r of any point P (x,y,z) is given as r = xî+yĵ+zk̂ , conversely if position vector r is given as r = xî+yĵ+zk̂ then , it’s point of coordinate is P(x,y,z). 2). The distance of any point P (x,y,z) from the origin O is given as √(x^2 + y^2 + z^2). It is equivalent to the |OP| vector. |

#### DISTANCE OF A POINT FROM THE COORDINATE AXES

#### SECTION FORMULA

## VECTOR FORM

#### INTERNAL DIVISION

#### EXTERNAL DIVISION

#### MID POINT

### COORDINATES OF THE GENERAL POINT

### CENTROID OF A TRIANGLE

### CENTROID OF A TETRAHEDRON

## DIRECTION COSINES

#### DIRECTION RATIOS

#### USEFUL RESULTS ON DIRECTION COSINES AND DIRECTION RATIOS

## DANGER |

1). The direction cosines of a line is unique but it’s direction ratios need not to be unique, it can be infinite. |

### ANGLES BETWEEN TWO LINES

## POINTS TO BE REMEMBERED |

1). If l1l2+m1m2+n1n2 = 0 then , the two vectors r1 and r2 having direction cosines l1,m1,n1 and l1,m2,n2 are orthogonal. 2). If l1/l2 = m1/m2 = n1/n2 are equal then the two vectors are parallel. 3). If a vector is equally inclined to the axes then the direction cosines are 4). If a line make α , β , γ and δ with the four diagonal of the cube then. 5). Angle between the any two diagonal of cube is cos-1(1/3). 6). Angle between the diagonal of the cube and the diagonal of the face is cos-1[√(2/3)]. |

### ANGLES IN TERMS OF DIRECTION RATIOS

## POINTS TO BE REMEMBERED |

If two lines are perpendicular then a1a2+b1b2+c1c2 = 0 , and if they are parallel then, a1/a2=b1/b2=c1/c2 |

### PROJECTION

## STRAIGHT LINES

### EQUATION OF THE LINE PASSING THROUGH A POINT AND PARALLEL TO A GIVEN VECTOR

#### VECTOR FORM

#### CARTESIAN FORM

## POINTS TO BE REMEMBERED |

1). The position vector of any point on the line is taken as = a+λb 2). If r is a position vector of any point P (x,y,z) then, r = xî+yĵ+zk̂ 3). Equation of the line whose direction cosines are l,m and n and passing through a fixed point (x1,y1,z1) is – (x-x1)/l =(y-y1)/m = (z-z1)/n 4). The coordinates of any point on the line (x-x1)/a = (y-y1)/b = (z-z1)/c is given as 5). The equation of X axes is *. Equation of y axes is *. Equation of z axes is |

### EQUATION OF THE LINE PASSING THROUGH TWO POINTS

#### VECTOR FORM

#### CARTESIAN FORM

### CHANGING UNSYMMETRICAL FORM TO SYMMETRICAL FORM

### ANGLES BETWEEN THE LINES

#### VECTOR FORM

#### CARTESIAN FORM

### INTERSECTION OF TWO LINES

#### TO FIND THE POINT OF INTERSECTION

__STEP(1):__write the coordinates of the general points of both lines equation as follow-

__STEP(2):__If these two lines Intersect then it must be a common point.

__STEP(3):__solve any two equation to find the value of λ and μ , after finding the value of λ and μ put its value into the coordinates of the general points in step 1.

### PERPENDICULAR FROM A POINT TO A LINE

#### VECTOR FORM

## SKEW LINES

### SHORTEST DISTANCE

### CALCULATION OF SHORTEST DISTANCE

#### VECTOR FORM

#### CARTESIAN FORM

## POINTS TO BE REMEMBERED |

1). Shortest distance between the two parallel lines r = a1+λb and r = a2+μb is given by – d = |(a2-a1)×b2|/|b|2). If the two lines r = a1+λb1 and r = a2+μb2 get Intersect , means d = 0 , then (a2-a1).(b1×b2) = 0 |

Well researchedwork!!