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.
|Text Book||NCERT Based|
|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
- The cartesian equation of a line through a point (x1, y1, z1) and having direction ratios a, b, c is (Symmetric form of a line).
- the coordinates of an arbitrary point on this line arc: (x1+λa, y1+ λb, z1+ 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 (x1, y1, z1) and (x2, y2, z2) is
- If θ is the acute angle between
and , then
- If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the acute angle between the two lines, then
- 1f the lines are perpendicular, then a1 a2 + b1 b2 + c1 c2 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
- Distance between parallel lines and
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 (x1,y1, z1) is a(x – x1) + b(y – y1) + c(z – z1) = O where a, b, c are direction ratios of the normal to the plane.
- Plane passing through three non-collinear points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) 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 A1x + B1y + C1z + D1 = O and A2x + B2y + C2z + D2 = O are cartesian equations of two planes inclined at an angle
The planes are parallel if
The planes are perpendicular if A1A2 + B1B2 + C1C2 = 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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