Plus Two Maths Notes Chapter 11 Three Dimensional Geometry is part of Plus Two Maths Notes. Here we have given Plus Two Maths Notes Chapter 11 Three Dimensional Geometry.

Board | SCERT, Kerala |

Text Book | NCERT Based |

Class | Plus Two |

Subject | Maths Notes |

Chapter | Chapter 11 |

Chapter Name | Three Dimensional Geometry |

Category | Plus Two Kerala |

## Kerala Plus Two Maths Notes Chapter 11 Three Dimensional Geometry

**Straight Lines In Space**

- Direction ratios of a line are the numbers which are proportional to the direction cosines of a line.
- It I, ni, n are the direction cosines and a, b, ç are the direction ratios of a line, then

- Vector equation of a line that passes through the given point whose position vector is and parallels to a given vector is = + λ; where λ is a

parameter. - The cartesian equation of a line through a point (x
_{1}, y_{1}, z_{1}) and having direction ratios a, b, c is (Symmetric form of a line). - the coordinates of an arbitrary point on this line arc: (x
_{1}+λa, y_{1}+ λb, z_{1}+ λc), where λ is a parameter. - Vector equation of a line that passes through two given points whose position vectors are and is = + λ( – ), where λ is a parameter.
- Cartesian equation of a line that passes through two points (x
_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is

- If θ is the acute angle between

and , then - If a
_{1}, b_{1}, c_{1}and a_{2}, b_{2}, c_{2}are the direction ratios of two lines and θ is the acute angle between the two lines, then

- 1f the lines are perpendicular, then a
_{1 }a_{2}+ b_{1 }b_{2}+ c_{1 }c_{2}O - If the lines are parallel, then

- Skew Lines are lines in space which are neither parallel nor intersecting. They lie in different planes.
- Shortest distance d between two skew lines

- In cartesian form

and - Distance between parallel lines and

is

**Planes in Space**

- Vector equation of a plane which is at a distance d from the origin, and is the unit vector normal to the plane through the origin is
- Cartesian equation of a plane which is at a distance of d from the origin and the direction cosines of the normal to the plane

as l, m, n is lx + my + nz = d - General equation of a plane is ax + by cz + d O, where a, b, c are direction ratios of normal to the plane.
- The equation of a plane through a point whose position vector is and perpendicular to the vector is
- The cartesian equation of a plane passing

through a given point (x_{1},y_{1}, z_{1}) is a(x – x_{1}) + b(y – y_{1}) + c(z – z_{1}) = O where a, b, c are direction ratios of the normal to the plane. - Plane passing through three non-collinear points (x
_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is

- Intercept form of equation of a plane is where a, b, c are intercepting lengths with X, Y and Z axes
- Vector equation of a plane that passes through the intersection of planes and where λ is any non – zero constant.
- In cartesian form

- If two planes are coplanar if

in cartesian form

- The angle between two planes is defined as the angle between their normals.

If and are two planes inclined at an angle θ, then

- If A
_{1}x + B_{1}y + C_{1}z + D_{1 }= O and A_{2}x + B_{2}y + C_{2}z + D_{2}= O are cartesian equations of two planes inclined at an angle

θ. then

The planes are parallel if

The planes are perpendicular if A_{1}A_{2}+ B_{1}B_{2}+ C_{1}C_{2}= 0 - The distance of a point whose position vector is a from the plane
- In cartesian

- The angle ϕ between a line and a plane is

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