12 class Maths Notes Chapter 11 Three Dimensional Geometry free PDF| Quick revision Three Dimensional Geometry Notes class 12 maths
CBSE Revision Notes for CBSE Class 12 Mathematics Three Dimensional Geometry Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.
Class 12 Maths Chapter-11 Three Dimensional Geometry Quick Revision Notes Free Pdf
CBSE Revision Notes for CBSE Class 12 Mathematics Three Dimensional Geometry Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.
Class 12 Maths Chapter-11 Three Dimensional Geometry Quick Revision Notes Free Pdf
🔷 Chapter - 11 🔷
👉 Three Dimensional Geometry 👈
1. CENTRAL IDEA OF 3D
There are infinite number of points in space. We want to identify each and every point of space with the help of three mutually perpendicular coordinate axes OX, OY and OZ.
2. AXES
Three mutually perpendicular lines OX, OY, OZ are considered as three axes.
3. COORDINATE PLANES
Planes formed with the help of x and y axes is known as x-y plane similarly y and z axes y – z plane and with z and x axis z - x plane.
4. COORDINATE OF A POINT
Consider any point P on the space drop a perpendicular form that point to x - y plane then the algebraic length of this perpendicular is considered as z-coordinate and from foot of the perpendicular drop perpendiculars to x and y axes these algebric length of perpendiculars are considered as y and x coordinates respectively.
5. VECTOR REPRESENTATION OF A POINT IN SPACE
If coordinate of a point P in space is (x, y, z) then the position vector of the point P with respect to the same origin is
6. DISTANCE FORMULA
7. DISTANCE OF A POINT P FROM COORDINATE AXES
Let PA, PB and PC are distances of the point P(x, y, z) from the coordinate axes OX, OY and OZ respectively then
8. SECTION FORMULA
(i) Internal Division :
If point P divides the distance between the points A (x₁, y₁, z₁,) and B (x₂, y₂, Z₂,) in the ratio of m: n (internally). The coordinate of P is given as
Note:
All these formulae are very much similar to two dimension coordinate geometry.
9. CENTROID OF A TRIANGLE
10. IN CENTRE OF TRIANGLE ABC
11. CENTROID OF A TETRAHEDRON
12. RELATION BETWEEN TWO LINES
Two lines in the space may be coplanar and may be none coplanar. Non coplanar lines are called skew lines if they never intersect each other. Two parallel lines are also non intersecting lines but they are coplanar. Two lines whether intersecting or non intersecting, the angle between them can be obtained.
(i) 13. DIRECTION COSINES AND DIRECTION RATIOS
Direction cosines : Let a, ß, y be the angles which directed line makes with the positive directions of the axes of x, y and z respectively, the cos a, cosß, cosy are called the direction cosines of the line. The direction cosine denoted (l, m, n).
(ii) If l, m, n, be the direction cosines of a lines, then l²+ m² + n² = 1
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines, l, m, n, then a, b, c are called the direction ratios.
If a, b, c are the direction ratio of any line L the
ai + bj+ ck will be a vector parallel to the line L.
If l, m, n are direction cosine of line L then i+mj+ nk is a unit vector parallel to the line L.
(iv) If l, m, n be the direction cosines and a, b, c be the direction ratios of a vector, then
(v) If OP = r, when O is the origin and the direction cosines of OP are l, m, n then the coordinates of P are (lr, mr, nr).
If direction cosine of the line AB are l, m, n, | AB |= r, and the coordinate of A is (x₁, y₁, z₁,) then the coordinate of B is given as (x₁, + rl, y₁, + rm, z₁ + rn)
(vi) If the coordinates P and Q are (x₁, y₁, Z₁) and (X₂, Y₂, Z₂) then the direction ratios of line PQ are, a = x₂, X₁, b = y₂ - y₁ and c = z₂ - z₁, and the direction cosines of
(vii) Direction cosines of axes : Since the positive x-axis makes angles 0°, 90°, 90° with axes of x, y and z respectively. Therefore
Direction cosines of x-axis are (1,0, 0)
Directio cosines of y-axis are (0, 1, 0)
Direction cosines of z-axis are (0, 0, 1)
14. ANGLE BETWEEN TWO LINE SEGMENTS
If two lines having direction ratios a₁, b₁, c₁, and a₂, b₂, c₂, respectively then we can consider two vector parallel to the lines as a,i + b,j+c,kand a₂i+b₂j+c₂k and angle between them can be given as.
15. PROJECTION OF A LINE SEGMENT ON A LINE
(i) If the coordinates P and Q are (x₁, Y₁, Z₁,) and (X₂, Y₂, Z₂) then the projection of the line segments PQ on a line having direction cosines l, m, n is
|l(x₂ - x₁) + m(y₂ - Y₁) + n(z₂ - z₁)I
A PLANE
If line joining any two points on a surface lies completely on it then the surface is a plane.
OR
If line joining any two points on a surface is perpendicular to some fixed straight line. Then this surface is called a plane. This fixed line is called the normal to the plane.
16. EQUATION OF A PLANE
(i) Normal form of the equation of a plane is lx + my + nz =p, where, l, mn are the direction cosines of the normal to the plane and p is the distance of the plane from the origin.
(ii) General form : ax + by + cz + d = 0 is the equation of a plane, where a, b, c are the direction ratios of the normal to the plane.
(iii) The equation of a plane passing through the point (x₁, Y₁, z₁) is given by a (x x₁)+ b(y-y₁) +c (z- z₁) = 0 where a, b, c are the direction ratios of the normal to the plane.
(iv) Plane through three points : The equation of the plane through three non-collinear points (x₁, Y₁, z₁),
Note.
(a) Vector equation of a plane normal to unitvector în and at a distance d from the origin is ř-n=d
(b) Planes parallel to the coordinate planes
(i) Equation of yz-plane is x = 0
(ii) Equation of xz-plane is y = 0
(iii) Equation of xy-plane is z=0
(c) Planes parallel to the axes :
If a = 0, the plane is parallel to x-axis i.e. equation of the plane parallel to the x-axis is by + cz + d = 0. Similarly, equation of planes parallel to y-axis and parallel to z-axis are ax + cz + d=0 and ax + by + d = 0 respectively.
(d) Plane through origin : Equation of plane passing through origin is ax + by + cz=0.
(e) Transformation of the equation of a plane to the normal form: To reduce any equation ax + by + cz-d=0 to the normal form, first write the constant term on the right hand side and make it positive, then divided each term
Where (+) sign is to be taken if d >0 an (-) sign is to be taken if d>0.
(F) Any plane parallel to the given plane ax + by + cz + d = 0 is ax + by + cz +λ = 0 distance between two parallel planes ax + by + cz + d₁ =0 and ax + dy + xz + d₂ =0
(h) A plane ax + by + cz + d = 0 divides the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂). in the ratio
17. ANGLE BETWEEN TWO PLANES
(i) Consider two planes ax + by + cz + d = 0 and a'x + b'y + c'z+ d' = 0. Angle between these planes is the angle between their normals. Since direction ratios of their normals are (a, b, c) and (a', b', c') respectively, hence Θ the angle between them is given by
18. A PLANE AND A POINT
(i) Distance of the point (x', y', z') from the plane
19. ANGLE BISECTORS
(i) The equations of the planes bisecting the angle between two given planes a₁x + b₁y + c₁z + d₁ =0 and a₂x + b₂y + c₂z + d₂ =0 are
(iii) Bisector of acute/obtuse angle : First make both the constant terms positive. Then
a₁a₂ + b₁b₂ + c₁c₂ >0
⇒ origin lies obtuse angle
a₁a₂ + b₁b₂ + c₁c₂ <0
⇒ origin lies in acute angle
20. FAMILY OF PLANES
(i) Any plane passing through the line of intersection of non- parallel planes or equation of the plane through the given line in non symmetrical form.
a₁x + b₁y + c₁z + d₁ =0 and a₂x + b₂y + c₂z + d₂ = 0 is a₁x + b₁y + c₁z+ d₁ + λ (a₂x + b₂y + c₂z + d₂) = 0
21. AREA OF A TRIANGLE
22. VOLUME OF A TETRAHEDRON
Volume of a tetrahedron with vertices A(x₁, y₁, z₁,), B(x₂, y₂, Z₂,), C(x₃, y₃, y₃) and D(x₄, y₄, y₄,) is given by
23. EQUATION OF A LINE
(i) A straight line in space is characterised by the intersection of two planes which are not parallel and therefore, the equation of a straight line is a solution of the system constituted by the equations of the two planes, a₁x + b₁y + c₁z +d₁ =0 and a₂x + b₂y + c₂z + d₂ = 0. This form is also known as non-symmetrical form.
(ii) The equation of a line passing through the point (X₁, y₁, Z₁,) and having direction ratios a, b, c is
24. ANGLE BETWEEN A PLANE AND A LIN
25. CONDITION FOR A LINE TO LIE IN A PLAN
26. COPLANER LINES
27. SKEW LINES
(i) The straight lines which are not parallel and non-coplanar ie. non-intersecting are called skew lines
28. COPLANARITY OF FOUR POINTS
The points A(x₁, y₁, z₁), B(x₂, y₂, Z₂) C(x₃, y₃, Z₃) and D(x₄, y₄, Z₄) are coplaner then
29. SIDES OF. PLANE
A plane divides the three dimensional space two equal parts. Two points A (x₁,y₁, z₁,) and B (x₂, y₂, z₂) are on the same side of the plane ax + by + cz + d 0 if ax₁ + by₁ + cz₁ + d and ax₂ + by₂ + cz₂ +d and both positive or both negative and are opposite side of plane if both of these values are in opposite sign.
30. LINE PASSING THROUGH THE GIVEN POINT (x₁ y₁ z₁) AND INTERSECTING BOTH THE LINES (P₁ = 0, P₂ = 0) AND (P₃ = 0, P₄ = 0)
Get a plane through (x₁, y₁, z₁) and containing the line (P₁ =0, P₂=0) as P₅=0
Also get a plane through (x₁, y₁, z₁) and containing the line P₃ = 0, P₄ = 0 as P₆ =0
equation of the required line is (P₅= 0, P₆ = 0)
31. TO FIND IMAGE OF A POINT W.R.T. A LINE
32. TO FIND IMAGE OF A POINT W.R.T. A PLANE
6. DISTANCE FORMULA
7. DISTANCE OF A POINT P FROM COORDINATE AXES
Let PA, PB and PC are distances of the point P(x, y, z) from the coordinate axes OX, OY and OZ respectively then
8. SECTION FORMULA
(i) Internal Division :
If point P divides the distance between the points A (x₁, y₁, z₁,) and B (x₂, y₂, Z₂,) in the ratio of m: n (internally). The coordinate of P is given as
Note:
All these formulae are very much similar to two dimension coordinate geometry.
9. CENTROID OF A TRIANGLE
10. IN CENTRE OF TRIANGLE ABC
11. CENTROID OF A TETRAHEDRON
12. RELATION BETWEEN TWO LINES
Two lines in the space may be coplanar and may be none coplanar. Non coplanar lines are called skew lines if they never intersect each other. Two parallel lines are also non intersecting lines but they are coplanar. Two lines whether intersecting or non intersecting, the angle between them can be obtained.
(i) 13. DIRECTION COSINES AND DIRECTION RATIOS
Direction cosines : Let a, ß, y be the angles which directed line makes with the positive directions of the axes of x, y and z respectively, the cos a, cosß, cosy are called the direction cosines of the line. The direction cosine denoted (l, m, n).
(ii) If l, m, n, be the direction cosines of a lines, then l²+ m² + n² = 1
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines, l, m, n, then a, b, c are called the direction ratios.
If a, b, c are the direction ratio of any line L the
ai + bj+ ck will be a vector parallel to the line L.
If l, m, n are direction cosine of line L then i+mj+ nk is a unit vector parallel to the line L.
(iv) If l, m, n be the direction cosines and a, b, c be the direction ratios of a vector, then
(v) If OP = r, when O is the origin and the direction cosines of OP are l, m, n then the coordinates of P are (lr, mr, nr).
If direction cosine of the line AB are l, m, n, | AB |= r, and the coordinate of A is (x₁, y₁, z₁,) then the coordinate of B is given as (x₁, + rl, y₁, + rm, z₁ + rn)
(vi) If the coordinates P and Q are (x₁, y₁, Z₁) and (X₂, Y₂, Z₂) then the direction ratios of line PQ are, a = x₂, X₁, b = y₂ - y₁ and c = z₂ - z₁, and the direction cosines of
(vii) Direction cosines of axes : Since the positive x-axis makes angles 0°, 90°, 90° with axes of x, y and z respectively. Therefore
Direction cosines of x-axis are (1,0, 0)
Directio cosines of y-axis are (0, 1, 0)
Direction cosines of z-axis are (0, 0, 1)
14. ANGLE BETWEEN TWO LINE SEGMENTS
If two lines having direction ratios a₁, b₁, c₁, and a₂, b₂, c₂, respectively then we can consider two vector parallel to the lines as a,i + b,j+c,kand a₂i+b₂j+c₂k and angle between them can be given as.
15. PROJECTION OF A LINE SEGMENT ON A LINE
(i) If the coordinates P and Q are (x₁, Y₁, Z₁,) and (X₂, Y₂, Z₂) then the projection of the line segments PQ on a line having direction cosines l, m, n is
|l(x₂ - x₁) + m(y₂ - Y₁) + n(z₂ - z₁)I
A PLANE
If line joining any two points on a surface lies completely on it then the surface is a plane.
OR
If line joining any two points on a surface is perpendicular to some fixed straight line. Then this surface is called a plane. This fixed line is called the normal to the plane.
16. EQUATION OF A PLANE
(i) Normal form of the equation of a plane is lx + my + nz =p, where, l, mn are the direction cosines of the normal to the plane and p is the distance of the plane from the origin.
(ii) General form : ax + by + cz + d = 0 is the equation of a plane, where a, b, c are the direction ratios of the normal to the plane.
(iii) The equation of a plane passing through the point (x₁, Y₁, z₁) is given by a (x x₁)+ b(y-y₁) +c (z- z₁) = 0 where a, b, c are the direction ratios of the normal to the plane.
(iv) Plane through three points : The equation of the plane through three non-collinear points (x₁, Y₁, z₁),
Note.
(a) Vector equation of a plane normal to unitvector în and at a distance d from the origin is ř-n=d
(b) Planes parallel to the coordinate planes
(i) Equation of yz-plane is x = 0
(ii) Equation of xz-plane is y = 0
(iii) Equation of xy-plane is z=0
(c) Planes parallel to the axes :
If a = 0, the plane is parallel to x-axis i.e. equation of the plane parallel to the x-axis is by + cz + d = 0. Similarly, equation of planes parallel to y-axis and parallel to z-axis are ax + cz + d=0 and ax + by + d = 0 respectively.
(d) Plane through origin : Equation of plane passing through origin is ax + by + cz=0.
(e) Transformation of the equation of a plane to the normal form: To reduce any equation ax + by + cz-d=0 to the normal form, first write the constant term on the right hand side and make it positive, then divided each term
Where (+) sign is to be taken if d >0 an (-) sign is to be taken if d>0.
(F) Any plane parallel to the given plane ax + by + cz + d = 0 is ax + by + cz +λ = 0 distance between two parallel planes ax + by + cz + d₁ =0 and ax + dy + xz + d₂ =0
(h) A plane ax + by + cz + d = 0 divides the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂). in the ratio
17. ANGLE BETWEEN TWO PLANES
(i) Consider two planes ax + by + cz + d = 0 and a'x + b'y + c'z+ d' = 0. Angle between these planes is the angle between their normals. Since direction ratios of their normals are (a, b, c) and (a', b', c') respectively, hence Θ the angle between them is given by
(i) Distance of the point (x', y', z') from the plane
19. ANGLE BISECTORS
(i) The equations of the planes bisecting the angle between two given planes a₁x + b₁y + c₁z + d₁ =0 and a₂x + b₂y + c₂z + d₂ =0 are
(iii) Bisector of acute/obtuse angle : First make both the constant terms positive. Then
a₁a₂ + b₁b₂ + c₁c₂ >0
⇒ origin lies obtuse angle
a₁a₂ + b₁b₂ + c₁c₂ <0
⇒ origin lies in acute angle
20. FAMILY OF PLANES
(i) Any plane passing through the line of intersection of non- parallel planes or equation of the plane through the given line in non symmetrical form.
a₁x + b₁y + c₁z + d₁ =0 and a₂x + b₂y + c₂z + d₂ = 0 is a₁x + b₁y + c₁z+ d₁ + λ (a₂x + b₂y + c₂z + d₂) = 0
21. AREA OF A TRIANGLE
22. VOLUME OF A TETRAHEDRON
Volume of a tetrahedron with vertices A(x₁, y₁, z₁,), B(x₂, y₂, Z₂,), C(x₃, y₃, y₃) and D(x₄, y₄, y₄,) is given by
23. EQUATION OF A LINE
(i) A straight line in space is characterised by the intersection of two planes which are not parallel and therefore, the equation of a straight line is a solution of the system constituted by the equations of the two planes, a₁x + b₁y + c₁z +d₁ =0 and a₂x + b₂y + c₂z + d₂ = 0. This form is also known as non-symmetrical form.
(ii) The equation of a line passing through the point (X₁, y₁, Z₁,) and having direction ratios a, b, c is
24. ANGLE BETWEEN A PLANE AND A LIN
25. CONDITION FOR A LINE TO LIE IN A PLAN
26. COPLANER LINES
27. SKEW LINES
(i) The straight lines which are not parallel and non-coplanar ie. non-intersecting are called skew lines
28. COPLANARITY OF FOUR POINTS
The points A(x₁, y₁, z₁), B(x₂, y₂, Z₂) C(x₃, y₃, Z₃) and D(x₄, y₄, Z₄) are coplaner then
29. SIDES OF. PLANE
A plane divides the three dimensional space two equal parts. Two points A (x₁,y₁, z₁,) and B (x₂, y₂, z₂) are on the same side of the plane ax + by + cz + d 0 if ax₁ + by₁ + cz₁ + d and ax₂ + by₂ + cz₂ +d and both positive or both negative and are opposite side of plane if both of these values are in opposite sign.
30. LINE PASSING THROUGH THE GIVEN POINT (x₁ y₁ z₁) AND INTERSECTING BOTH THE LINES (P₁ = 0, P₂ = 0) AND (P₃ = 0, P₄ = 0)
Get a plane through (x₁, y₁, z₁) and containing the line (P₁ =0, P₂=0) as P₅=0
Also get a plane through (x₁, y₁, z₁) and containing the line P₃ = 0, P₄ = 0 as P₆ =0
equation of the required line is (P₅= 0, P₆ = 0)
31. TO FIND IMAGE OF A POINT W.R.T. A LINE
32. TO FIND IMAGE OF A POINT W.R.T. A PLANE