Three Dimensional Geometry — Class 12 CBSE Mathematics revision notes
A 3-page visual revision sheet for Three Dimensional Geometry, written for Class 12 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Three Dimensional Geometry in one look
a line needs a direction and a point, a plane needs a normal and a distance
Direction cosines vs direction ratios
- cosines of the angles the line makes with the x, y, z axes
- always satisfy l^2+m^2+n^2=1
- unique for a given directed line, reversing the direction flips every sign
- any numbers proportional to l, m, n
- not unique: (2,4,6) and (1,2,3) describe the same direction
- recovered as l = a/√a^2+b^2+c^2, and likewise for m and n
Equation of a line
- vector form, through a point a, direction b
r = a + lambdab - cartesian form, through (x_1,y_1,z_1), direction ratios a,b,c
x-x_1/a = y-y_1/b = z-z_1/c - through two points
x-x_1/x_2-x_1 = y-y_1/y_2-y_1 = z-z_1/z_2-z_1
Angle between two lines
- direction ratios (a_1,b_1,c_1) and (a_2,b_2,c_2)
cosθ = frac|a_1a_2+b_1b_2+c_1c_2|√a_1^2+b_1^2+c_1^2√a_2^2+b_2^2+c_2^2 - perpendicular exactly when
a_1a_2+b_1b_2+c_1c_2 = 0 - parallel exactly when
a_1/a_2 = b_1/b_2 = c_1/c_2
Page 1 of 3. The full sheet, all 3 pages, opens free in the app.
What this Three Dimensional Geometry sheet covers
- Direction cosines vs direction ratios
- Direction cosines (l, m, n)
- Direction ratios (a, b, c)
- Equation of a line
- vector form, through a point a, direction b
- cartesian form, through (x_1,y_1,z_1), direction ratios a,b,c
- through two points
- Angle between two lines
- direction ratios (a_1,b_1,c_1) and (a_2,b_2,c_2)
- perpendicular exactly when
- parallel exactly when
- Two lines, three possible relationships
- Equation of a plane
- normal form, unit normal n-cap, distance d from the origin
- cartesian normal form, l,m,n direction cosines of the normal
- through a point, normal direction ratios a,b,c
- intercept form, axis intercepts a,b,c
- Angle between planes, and between a line and a plane
- between two planes, normals N1 and N2
- perpendicular planes
- between a line (direction b) and a plane (normal N)
- Shortest distance between two skew lines
- Classifying two lines in space
- Distance formulas at a glance
- Things to remember before the traps
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Three Dimensional Geometry the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.