Three Dimensional Geometry

Formulas are organized based on their importance in the CAT exam.
1 Parallel Lines
2 Plane Through Origin
3 Acute Angle Between Two Lines
4 Direction Cosines of a Plane Normal
5 Relation Between Direction Cosines
6 Foot of perpendicular from point to plane
7 Vector Equation of a Line
8 Distance Between a Point and a Plane Through a Point
9 Length of Tangent from an External Point
10 Tangent Length and Power
11 Volume of a Sphere
12 Orthogonal Spheres
13 Line Parallel to a Plane
14 Distance Between Two Skew Lines Using Scalar Triple Product
15 Concentric Spheres
16 Distance Between Two Intersecting Lines
17 Angle between line and plane
18 Perpendicular bisector plane
19 Angle between two lines
20 Two Spheres Do Not Intersect
21 Parametric Equation of a Line
22 Plane Containing Two Parallel Lines
23 Radical Axis / Radical Plane Condition
24 Direction Ratios of Line Joining Two Points
25 Polar Plane of a Point on the Sphere
26 Surface Area of a Sphere
27 Distance Between Two Points in Vector Form
28 Midpoint of Two Points
29 Condition for Two Lines to Be Skew
30 Tangent Plane to Sphere in Centre-Radius Form
31 Length of Tangent from a Point to a Sphere
32 Line Through a Given Point Parallel to a Given Line
33 Radius from Diameter
34 Distance from point to plane
35 Tangent Plane Condition
36 Condition for Tangency of a Plane and Sphere
37 Distance from point to line
38 Two Spheres Touch Internally
39 Volume of Tetrahedron
40 Two Spheres Touch Externally
41 Cartesian Equation Through Two Points
42 Coplanarity of two lines
43 Volume of Parallelepiped
44 Plane Through Two Intersecting Lines
45 Radical Plane of Two Spheres
46 Distance from Origin
47 Sphere with Centre on the z-Axis
48 Distance Between Two Points and Midpoint Relation
49 Two Spheres Intersect in a Circle
50 Angle Bisector Plane
51 Equation of a Plane Parallel to a Given Plane
52 Line Perpendicular to a Plane
53 Condition for a Plane to Touch a Sphere
54 Pole-Polar Reciprocity
55 Projection on Coordinate Axes
56 General equation of a plane
57 Resolution of a Vector
58 Area of a Great Circle
59 Plane Through Three Points in Vector Form
60 Condition for a Plane Not to Meet a Sphere
61 Intersection of Two Spheres
62 Normal Vector to a Plane
63 Skew Lines
64 Midpoint formula
65 Distance from Origin to a Plane
66 Plane as Locus of Points
67 General Equation of a Sphere
68 Perpendicular Lines
69 Vector Equation of Line Through Two Points
70 Perpendicular Planes
71 Parallel Planes
72 Line Perpendicular to a Plane
73 Cartesian Equation of a Line
74 Distance Between Two Points
75 Direction Ratios of a Line
76 Direction Cosines from Direction Ratios
77 Direction Ratios from Direction Cosines
78 Condition for a Line to Be Parallel to a Plane
79 Direction Cosines of Line Joining Two Points
80 Angle Between Lines Using Direction Ratios
81 Distance of a Point from a Line
82 Plane Through Three Points
83 Equation of the Common Perpendicular
84 Plane Parallel to a Line
85 Condition for a Point to Lie Outside a Sphere
86 Sphere with Diameter as a Given Line Segment
87 Power of a Point with Respect to a Sphere
88 Plane Tangent to Sphere at a Given Point
89 Normal and Tangent Plane Relationship
90 Distance Between Parallel Planes in Vector Form
91 Foot of Perpendicular from a Point to a Plane
92 Reflection of a Point in a Plane
93 Foot of Perpendicular from a Point to a Line
94 Reflection of a Point in a Line
95 Distance Between Two Lines
96 Shortest Distance Between Two Lines Using Determinant
97 Condition for Two Lines to Be Parallel
98 Plane Through a Line and Perpendicular to Another Plane
99 Plane Through a Line and Perpendicular to Another Line
100 Tangent Plane from a Point to a Sphere
101 Circle of Intersection of a Sphere and a Plane
102 Tangent Plane as Limiting Secant Plane
103 Sphere Touching a Line
104 Line Tangent to a Sphere
105 Plane Tangent to a Sphere at a Point
106 Normal to a Sphere at a Point
107 Tangent Plane to General Sphere
108 Distance Between Centres of Two Spheres
109 Radical Plane of Two Spheres in General Form
110 Common Chord of Two Spheres
111 Centre of the Circle of Intersection
112 Locus of Points at a Fixed Distance from a Point
113 Locus of Points Equidistant from Two Points
114 Direction Ratios of a Plane Normal
115 Direction Ratios of Intersection of Two Planes
116 Perpendicular Distance Between a Point and a Line
117 Vector Equation of a Plane
118 Intercept Form of a Plane
119 Equation of a Line Through a Point and Parallel to a Vector
120 Cartesian Form from Vector Form
121 Vector Form from Cartesian Line
122 Condition for Two Planes to Intersec
123 Three-Dimensional Coordinate System
124 Distance of a Point from a Plane
125 Distance Between Two Parallel Planes
126 Angle Between Two Planes
127 Angle Between a Line and a Plane
128 Line of Intersection of Two Planes
129 Condition for Parallel Distinct Planes
130 Plane Through a Given Line
131 Equation of a Plane Perpendicular to a Given Plane
132 Mid-Plane Between Two Parallel Planes
133 Distance Ratio of a Point from Two Parallel Planes
134 Plane Through the Intersection of Two Planes
135 Family of Planes Through a Given Line
136 Chord of Contact
137 Polar Plane of a Point
138 Chord of Contact as Polar Plane
139 Pole of a Plane
140 Equation of a Sphere with Centre on a Given Line
141 Equation of a Circle in a Plane
142 Circle as Intersection of Sphere and Plane
143 Radius of Circle of Intersection
144 Angle between two planes
145 Centroid of a Tetrahedron
146 Centroid of a Tetrahedron in Coordinates
147 Collinearity of Three Points
148 Coplanarity of Four Points Using Determinant
149 Plane Through Three Non-Collinear Points
150 Sphere Passing Through the Origin and a Given Point
151 Intersection of a Sphere and a Coordinate Plane
152 Shortest Distance Between Two Skew Lines
153 Distance between two points
154 Point on a Sphere
155 Common Perpendicular to Two Skew Lines
156 Direction Cosines of a Line
157 Orthogonal Spheres
158 Angle Between Two Lines Using Direction Vectors
159 Angle Between Two Planes Using Normal Vectors
160 Sphere with Centre on the yz-Plane
161 Sphere with Centre on the zx-Plane
162 Normal Form of a Plane
163 Plane Through a Given Point Parallel to a Given Plane
164 Plane Through a Given Point Perpendicular to a Given Line
165 Plane Bisecting Two Intersecting Planes
166 Plane Equidistant from Two Parallel Planes
167 Polar Plane of a Point for Sphere with Centre at Origin
168 Section Formula in 3D
169 External Section Formula in 3D
170 Area of Triangle in 3D
171 Normal to a Sphere
172 Radius of Common Circle
173 Shortest distance between skew lines
174 Coplanarity of Four Points
175 Distance of a Point from Two Planes
176 Locus of a Point Equidistant from Two Planes
177 Position Vector of a Point
178 Sphere with Centre at Origin
179 Volume of Tetrahedron
180 Equation of a Sphere
181 Centre of a Sphere from General Equation
182 Condition for a Plane to Intersect a Sphere
183 Sphere with Centre on a Coordinate Plane
184 Locus of Points Equidistant from Two Intersecting Planes
185 Locus of Points Equidistant from Two Parallel Planes
186 Normal Vector to Plane Through Three Points
187 Coordinate Planes
188 Angle Between a Line and a Plane Using Direction Cosines
189 Condition for a Line to Be Perpendicular to a Plane
190 Distance Between a Point and a Line Using Cross Product
191 Equation of Normal to a Sphere
192 Plane Through the Centre of a Sphere
193 Plane Through a Given Point Parallel to a Given Line
194 Angle Between a Line and a Plane
195 Condition for Line Perpendicular to Plane
196 Projection of a Vector on Another Vector
197 Condition for Two Lines to Be Perpendicular
198 Projection of a Point on a Plane
199 Reflection of a Point in a Plane Using Vectors
200 Sphere Tangent to a Coordinate Plane
201 Sphere Tangent to a Coordinate Axis
202 Sphere Tangent to a Plane
203 Sphere Tangent to Two Parallel Planes
204 Centre of a Sphere Tangent to a Plane
205 Normal to General Sphere
206 Direction cosines and ratios
207 Angle Between Two Lines
208 Distance Between a Point and a Plane in Vector Form
209 Coplanarity of Two Lines
210 Intersection of Two Lines
211 Volume of Tetrahedron from Four Points
212 Radius of a Sphere from General Equation
213 Plane Perpendicular to a Line
214 Plane through point with normal vector
215 Intersection of a Line and a Plane
216 Condition for Line Parallel to Plane
217 General Equation of a Plane from Normal and Point
218 Equation of Plane Through Origin and Two Direction Vectors
219 Distance Ratio Along a Line
220 Foot of Perpendicular from a Point to a Line
221 Scalar Triple Product and Volume
222 Sphere Passing Through the Origin
223 Direction Vector of the Line of Intersection
224 Condition for Coincident Planes
225 Sphere Through Four Points
226 Diameter Form of a Sphere
227 Centre of Sphere from Diameter
228 Tangent Plane to a Sphere
229 Vector Projection
230 Perpendicular Component of a Vector
231 Sphere Through a Given Point
232 Sphere Tangent to a Plane and Passing Through a Point
233 Area of Cross-Section of a Sphere
234 Great Circle
235 Radius of Cross-Section of a Sphere
236 Distance of Radical Plane from First Centre
237 Midpoint Plane for Equal-Radius Spheres
238 Condition for a Real Sphere
239 Coplanarity of Two Lines
240 Condition for a Point to Lie on a Sphere
241 Line Lies in a Plane
242 Equations of a line
243 Symmetric to Parametric Form
244 Condition for a Point to Lie Inside a Sphere
245 Shortest Distance Between Two Parallel Lines
246 Final Coordinate Geometry Identity
247 Plane Containing Two Lines
248 General Equation of a Plane
249 Plane Through a Line and a Point
250 Plane Through Two Parallel Lines
251 Midpoint of Two Points on a Line
252 Equation of a Plane
253 Radius of the Common Circle of Two Spheres
254 Sphere with Centre on a Coordinate Axis
255 Sphere with Centre on the y-Axis
256 Distance Between Two Points on a Line
257 Line Through a Given Point Perpendicular to a Given Line
258 Plane Parallel to Two Lines
259 Plane Containing Two Intersecting Lines
260 Perpendicular Distance Between a Point and a Plane
261 Distance Between Parallel Lines
262 Distance Between Skew Lines
263 Distance Between Parallel Planes
264 Plane at a Given Distance from Origin
265 Line Parallel to a Plane
266 Line Lies in a Plane