With the help of 3D Geometry IIT JEE notes PDF, you can clear all your concepts related to this three-dimensional geometry.

By Mohit ChauhanPublished : Aug 6, 2020 , 13:38 IST 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(x1, y1, z1) and Q(x2, y2, z2) in the three-dimensional Cartesian system is given by ## 2. Section formula

If C(x, y, z) divides the join of A(x1, y1, z1) and B(x2, y2, z2) in the ratio m1 : m2(m1, m2 > 0) then and ## 3. Centroid of a Triangle

The centroid of a triangle XYZ whose vertices are X(x1, y1, z1), Y(x2, y2, z2) and Z(x3, y3, z3) are

## 4. Centroid of a Tetrahedron

The centroid of a tetrahedron ABCD whose vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4) 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) 2 + m2 + n2 = 1 or cos2α + cos2β + cos2γ = 1

(c) Sin2α + sin2β + sin2γ =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 (x2, y2, z2) respectively, are x2 – x1, y2 – y1, z2 – z1.

## 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 (ℓ1, m1, n1) and (ℓ2, m2, n2) be the direction cosines of any two lines and θ be the angle between them, then, cos θ = ℓ12 + m1m2 + n1n2

(a) If lines are perpendicular then l1 l2 + m1m2 + n1n2 = 0

(b) If lines are parallel then (c) If the d.r.’s of the two lines are a1, b1, c1 and a2, b2, c2 then & So, that the conditions for perpendicular and parallelism of two lines are repetitively.

a1a2 + b1b2 + c1c2 = 0 and (d) If l1, m1, n1 & ℓ2, m2, n2  are the d.c.’s of two lines, the d.r.’s of the line which is perpendicular to both of them are  m1n2 – m2n1,  n1l1 – n2 l2, l1m2 – l2m1.

## 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 (x1, y1, z1) is

a(x – x1) + b(y – y1) + c(z – z1) = 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(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) 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(x1, y1, z1) and parallel to two lines whose d.c’s and l1, m1, n1 and l2, m2, n2 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(x1, y1, z1) and Q(x2, y2, z2) 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 a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 then (a)If planes are perpendicular then a1a2 + b1b2 + c1c2 = 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 (x1, y1, z1) on ax + by + cz + d = 0 is ## 13. Positions of Points (x1, y1, z1) and (x2, y2, z2) relative to a Plane

If the points (x1, y1, z1) and (x2, y2, z2) 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 + d1 = 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 a1x + b1y + c1z + d1 = 0 can be represented by the equation.

(ax = by + cz + d) + λ (a1x + b1y + c1z + d1) = 0

## 16. Equations of Bisectors of the Angles between two Planes

Equations of the bisectors of the planes

P1 : ax + by + cz + d = 0 & P2 : a1x + b1y + c1z + d1 = 0

(where d > 0 & d1 >0) are Conditions Acute angel Bisector Obtuse angle Bisector aa1 + bb1 + cc1 > 0 – + aa1 + bb1 + cc1 < 0 + –

## 17. The Image of a Point with respect to Plane Mirror

The image of A(x1, y1, z1) with respect to the plane mirror ax + by + cz + d = 0 be B(x2, y2, z2) is given by

## 18. The feet of the perpendicular from a point on a plane

The feet of perpendicular from a point A(x1, y1, z1) on the ax + by + cz + d = 0 be B(x2, y2, z2) is given by ## 19. Reflection of a plane on another plane

The reflection of the plane ax + by + cz + d = 0 on the place a1x + b1y + c1z + d1 = 0 is

2(aa1 + bb1 + cc1), (a1x + b1y + c1z + d1)= (a12 + b12 + c12 )(ax+by+cz+d)

## 20. Area of a Triangle

If Ayz, Azx, Axy be the projections of an area A on the co-ordinate planes yz, zx and xy respectively, then If vertices of a triangle are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) then Note The area of triangle = ½ bc sin A.

## 21. Pair of Planes: Homogeneous Equation of Second degree

An equation of the form ax2 + by2 + cz2 + 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 – af2 – bg2 – ch2 = 0

## 22. The angle between two Planes

If θ is the acute angle between two planes whose joint equation is ax2 + by2 + cz2 + 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 a1x + b1y + c1z + d1 = 0 be the equations of any two planes, taken together then ax + by + cz + d = 0 = a1x + b1y + c1z + d1 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 (x1, y1, z1) 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 (x1 + ar, y1 + br, z1 + cr) when r is directed distance.

## 25. The equation of line Passing through two Points

The equations of the line passing through (x1, y1, z1) and (x2, y2, z2) is ## 26. Symmetric Form of the equation of the line

The equation of the line passing through (x1, y1, z1) 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 a1x + b1y + c1z + 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 aa1 + bb1 + cc1 = 0

## 28. General Equation of the Plane containing the Line is a(x – x1) + b(y – y1) + c(z – z1) where al + bm + cn = 0

## 29. Coplanar lines

### (i) Equations of both lines in the symmetrical form

If the two lines are coplanar then & the equation  of the plane containing the line is

### (ii)If one line in the symmetrical form

Let lines are and a1x + b1y + c1z + d1 = 0 = a2x + b2y + c2z + d2

The condition for coplanarity is ### (iii) If both lines are in General form

Let lines are a1x + b1y + c1z + d1 = 0 = a2x + b2y + c2z + d2

and  a3x + b3y + c3z + d3 = 0 = a4x + b4y + c4z + d4

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(x2 – x1) + m (y2 – y1) + n(z2 – z1)|

and

Equation of the shortest distance is and ## 31. Volume of Tetrahedron

If vertices of the tetrahedron are (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4) is More from us:

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