3D Geometry Notes for IIT JEE, Download PDF
With the help of 3D Geometry IIT JEE notes PDF, you can clear all your concepts related to this three-dimensional geometry.
Three-dimensional geometry is one of the most interesting topics to study in mathematics. For a proper understanding of the topic practice previous year questions. It comprises 2-3 questions in JEE Main/ JEE Advanced and other engineering entrance examinations. Download the pdf of the 3D Geometry notes from the link given at the end of the article.
1. The distance between two points in three dimensions cartesian system
The distance between two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) in the three-dimensional Cartesian system is given by
2. Section formula
If C(x, y, z) divides the join of A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) in the ratio m_{1} : m_{2}(m_{1}, m_{2} > 0) then
and
3. Centroid of a Triangle
The centroid of a triangle XYZ whose vertices are X(x_{1}, y_{1}, z_{1}), Y(x_{2}, y_{2}, z_{2}) and Z(x_{3}, y_{3}, z_{3}) are
4. Centroid of a Tetrahedron
The centroid of a tetrahedron ABCD whose vertices A(x_{1}, y_{1}, z_{1}), B(x_{2}, y_{2}, z_{2}), C(x_{3}, y_{3}, z_{3}) and D(x_{4}, y_{4}, z_{4}) are
5. Direction Cosines (d.c’s)
If a line makes in angles α, β, γ with positive directions of x, y and z-axes then cos α, cos β, cos γ are called the direction cosines of the line. Generally, direction cosines are represented by l, m, n.
The angle α, β, γ are called the direction angles of the line XY and the direction cosines of YX are cos(π – α), cos(π – γ) i.e., – cos α, –cos β, –cos γ.
Thus, the direction cosines of the x-axis are cos 0, cos π/2, cos π/2 i.e., 1, 0, 0. Similarly the d.c’s of y and z axis are (0, 1, 0) and (0, 0, 1) respectively.
Note:
(a) If ℓ, m, n be the d.c’s of a line OP and OP = r, then the coordinates of the point P are (ℓr, mr, nr).
(b) ℓ^{2} + m^{2} + n^{2} = 1 or cos^{2}α + cos^{2}β + cos^{2}γ = 1
(c) Sin^{2}α + sin^{2}β + sin^{2}γ =2
6. Direction ratios (d.r.’s)
Direction ratios of a line are numbers which are proportional of the d.c’s of a line.
Direction ratios of a line PQ, (where P and Q are (x1, y1, z1) and (x_{2}, y_{2}, z_{2}) respectively, are x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1}.
7. The relation between the d.c.’s and d.r.’s
If a, b, c and the d.r.’s and l, m, n are the d.c.’s, then
Remember: If a, b, c and d.r.’s of AB then d.c’s of a line AB are given by the +ve sign and those of the line BA by –ve sign.
8. The angle between the two lines
If (ℓ_{1}, m_{1}, n_{1}) and (ℓ_{2}, m_{2}, n_{2}) be the direction cosines of any two lines and θ be the angle between them, then, cos θ = ℓ_{1} ℓ_{2} + m_{1}m_{2} + n_{1}n_{2}
(a) If lines are perpendicular then l_{1} l_{2} + m_{1}m_{2} + n_{1}n_{2 }= 0
(b) If lines are parallel then
(c) If the d.r.’s of the two lines are a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} then
&
So, that the conditions for perpendicular and parallelism of two lines are repetitively.
a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0 and
(d) If l_{1}, m_{1}, n_{1} & ℓ_{2}, m_{2}, n_{2 } are the d.c.’s of two lines, the d.r.’s of the line which is perpendicular to both of them are m_{1}n_{2} – m_{2}n_{1}, n_{1}l_{1} – n_{2 }l_{2}, l_{1}m_{2 }– l_{2}m_{1}.
9. The equation of a plane in three-dimensional geometry
(i) General form
Every equation of first degree in x, y, z represents a plane. The most general equation of the first degree in x, y, z is ax + by + cz + d = 0 where at least one of a,b,c is non-zero.
Note:
(a) Equation of yz plane is x = 0
(b) Equation of zx plane is y = 0
(c) Equation of xy plane is z = 0
(ii) One-point form
The equation of the plane through (x_{1}, y_{1}, z_{1}) is
a(x – x_{1}) + b(y – y_{1}) + c(z – z_{1}) = 0
(iii) Intercept from
The equation of the plane in terms of intercepts of a,b,c from the axes is
(iv) Normal form
The equation of plane on which the perpendicular from the origin of length p and the direction cosines of perpendicular are cos α, cosβ and cos γ with the positive directions of x, y & z-axes respectively is given by x cos α + y cos β + z cos γ = p
(v) The equation of the plane passing through three given points
The equation of the plane passing through A(x_{1}, y_{1}, z_{1}), B(x_{2}, y_{2}, z_{2}) and C(x_{3}, y_{3}, z_{3}) is given by
(vi) The equation of a plane passing through a point and parallel to two lines
The equation of the plane passing through a point P(x_{1}, y_{1}, z_{1}) and parallel to two lines whose d.c’s and l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} is
(vii) The equation of a plane passing through two points and parallel to all line
The equation of the plane passes through two point P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) and is parallel to a line whose d.c.’s are l, m, n is
10. Angle Between two Planes
If θ be the angle between the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 then
(a)If planes are perpendicular then a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0
(b) If planes are parallel then
11. Angle Between a Plane and a Line
If α be the angle between the normal to the plane and a line then 90° – α is the angle between the plane and the line.
12. Length of Perpendicular from a Point to a Plane
The length of perpendicular from (x_{1}, y_{1}, z_{1}) on ax + by + cz + d = 0 is
13. Positions of Points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) relative to a Plane
If the points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) are on the same side or opposite side of the plane ax + by + cz + d = 0 then
14. The distance between the Parallel Planes
Let two parallel planes be ax + by + cz + d = 0 and ax + by + cz + d_{1} = 0
Direct Formula: The distance between parallel planes is
Alternate Method: Find the coordinates of any point on one of the given planes, preferably putting x = 0, y = 0 or y = 0 z = 0 or z = 0, x = 0. Then the perpendicular distance of this point from the other plane is the required distance between the planes.
15. Family of Planes
Any plane passing through the line of inter-section of the planes ax + by + cz + d = 0 and a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 can be represented by the equation.
(ax = by + cz + d) + λ (a_{1}x + b_{1}y + c_{1}z + d_{1}) = 0
16. Equations of Bisectors of the Angles between two Planes
Equations of the bisectors of the planes
P_{1} : ax + by + cz + d = 0 & P_{2} : a_{1}x + b_{1}y + c_{1}z + d_{1} = 0
(where d > 0 & d_{1} >0) are
Conditions |
Acute angel Bisector |
Obtuse angle Bisector |
aa_{1} + bb_{1} + cc_{1} > 0 |
– |
+ |
aa_{1} + bb_{1} + cc_{1} < 0 |
+ |
– |
17. The Image of a Point with respect to Plane Mirror
The image of A(x_{1}, y_{1}, z_{1}) with respect to the plane mirror ax + by + cz + d = 0 be B(x_{2}, y_{2}, z_{2}) is given by
18. The feet of the perpendicular from a point on a plane
The feet of perpendicular from a point A(x_{1}, y_{1}, z_{1}) on the ax + by + cz + d = 0 be B(x_{2}, y_{2}, z_{2}) is given by
19. Reflection of a plane on another plane
The reflection of the plane ax + by + cz + d = 0 on the place a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 is
2(aa_{1} + bb_{1} + cc_{1}), (a_{1}x + b_{1}y + c_{1}z + d_{1})= (a_{1}^{2} + b_{1}^{2} + c_{1}^{2} )(ax+by+cz+d)
20. Area of a Triangle
If A_{yz}, A_{zx}, A_{xy} be the projections of an area A on the co-ordinate planes yz, zx and xy respectively, then
If vertices of a triangle are (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) then
Note The area of triangle = ½ bc sin A.
21. Pair of Planes: Homogeneous Equation of Second degree
An equation of the form ax^{2} + by^{2} + cz^{2} + 2fyz + 2gzx + 2hxy = 0 is called a homogeneous equation of second degree. It represents two planes passing through the origin. The condition that it represents a plane is
abc + 2fgh – af^{2} – bg^{2} – ch^{2} = 0
22. The angle between two Planes
If θ is the acute angle between two planes whose joint equation is ax^{2} + by^{2} + cz^{2} + 2fyz + 2gzx + 2hxy = 0, then
Note: If planes are perpendicular then a + b + c = 0
23. General equation of a straight line
Let ax + by + cz + d = 0 and a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 be the equations of any two planes, taken together then ax + by + cz + d = 0 = a_{1}x + b_{1}y + c_{1}z + d_{1} is the equation of straight line.
The x-axis has equations y = 0 = z, the y-axis z = 0 = x and the z-axis x = 0 = y.
24. The equation of a line Passing through a Point and Parallel to a Specified Direction
The equation of the line passing through (x_{1}, y_{1}, z_{1}) and parallel to a line whose d.r.’s an a, b, c is
and the co-ordinate of any point on the line an (x_{1} + ar, y_{1} + br, z_{1} + cr) when r is directed distance.
25. The equation of line Passing through two Points
The equations of the line passing through (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is
26. Symmetric Form of the equation of the line
The equation of the line passing through (x_{1}, y_{1}, z_{1}) and having direction cosines l, m, n is
27. To convert from General Equation of a line to Symmetrical Form
(i) Point: Put x = 0 (or y = 0 or z = 0) in the given equations and solve for y and z.
The values of x, y and z are the coordinates of a point lying on the line.
(ii) Direction cosines: Since the line is perpendicular to the normal to the given planes then find direction cosines. Then write the equation of the line with the help of a point & direction cosines.
28. The angle between a Line and a Plane
If the angle between the line and the plane a_{1}x + b_{1}y + c_{1}z + d = 0 is θ then 90° – θ is the angle between normal and the line
i.e.,
or
Note: If line is parallel to the plane then aa_{1} + bb_{1} + cc_{1} = 0
28. General Equation of the Plane containing the Line
is a(x – x_{1}) + b(y – y_{1}) + c(z – z_{1}) where al + bm + cn = 0
29. Coplanar lines
(i) Equations of both lines in the symmetrical form
If the two lines arecoplanar then
& the equation of the plane containing the line is
(ii) If one line in the symmetrical form
Let lines are and a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 = a_{2}x + b_{2}y + c_{2}z + d_{2}
The condition for coplanarity is
(iii) If both lines are in General form
Let lines are a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 = a_{2}x + b_{2}y + c_{2}z + d_{2 }
and a_{3}x + b_{3}y + c_{3}z + d_{3} = 0 = a_{4}x + b_{4}y + c_{4}z + d_{4}
The condition that this pair of liens is coplanar is
30. Shortest Distance
Two straight lines in space when they are not coplanar are called skew lines. Thus skew lines are neither parallel nor intersect at any point. Let PQ and RS are two skew lines and a line which is perpendicular to both PQ and RS. Then the length of the lines is called the shortest distance between PQ and RS.
Let equations of the given lines are
Let S.D. lie along the line
S.D. = |l(x_{2} – x_{1}) + m (y_{2} – y_{1}) + n(z_{2} – z_{1})|
and
Equation of the shortest distance is
and
31. Volume of Tetrahedron
If vertices of the tetrahedron are (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}) and (x_{4}, y_{4}, z_{4}) is
Three Dimensional Geometry Notes for IIT JEE, Download PDF
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