Knowledge organisers

Three Dimensional Geometry: CBSE Class 12 Maths knowledge organiser

Everything to know about three dimensional geometry on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Direction cosines: The cosines of the angles a line makes with the x-, y- and z-axes; l² + m² + n² = 1.
Direction ratios: Any numbers proportional to the direction cosines.
Skew lines: Lines in space that are neither parallel nor intersecting.

Key formulas

DCs from direction ratios \(a,b,c\)	\(l=\frac{a}{\sqrt{a^2+b^2+c^2}},\) \(\dots\)
Line through \(\vec a\) parallel to \(\vec b\)	\(\vec r=\vec a+\lambda\vec b;\) \(\frac{x-x_1}a=\frac{y-y_1}b=\frac{z-z_1}c\)
Through two points	\(\vec r=\vec a+\lambda(\vec b-\vec a);\) \(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\)
Angle between lines	\(\cos\theta=\frac{\|a_1a_2+b_1b_2+c_1c_2\|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\)
Perpendicular; parallel	\(a_1a_2+b_1b_2+c_1c_2=0;\) \(\tfrac{a_1}{a_2}=\tfrac{b_1}{b_2}=\tfrac{c_1}{c_2}\)
Shortest distance, skew lines	\(d=\frac{\big\|(\vec b_1\times\vec b_2)\cdot(\vec a_2-\vec a_1)\big\|}{\|\vec b_1\times\vec b_2\|}\)
Parallel lines	\(d=\frac{\|\vec b\times(\vec a_2-\vec a_1)\|}{\|\vec b\|}\)
Lines intersect ⇔ shortest distance = 0. Foot of perpendicular from \(P\): general point \(Q\) on the line with \(\overrightarrow{PQ}\cdot\vec b=0\)

Worked example

Line through \((1, 0, -1)\) and \((3, 2, 0)\): \(\dfrac{x - 1}{2} = \dfrac{y}{2} = \dfrac{z + 1}{1}\); direction cosines \(\tfrac23, \tfrac23, \tfrac13\).

Common mistakes

Reading direction ratios before putting the line in standard form (signs and coefficients of \(x, y, z\)).
Omitting the modulus in the angle formula (the angle between lines is taken acute) or in the distance formula.
Using the skew-line formula for parallel lines (it gives \(0/0\)).
Declaring lines intersecting after matching only two coordinates — always check the third.

You should be able to…

Find direction cosines/ratios and write vector and Cartesian equations of a line through a point or two points.
Convert between forms, find angles between lines, test parallel/perpendicular/coincident lines and distances between parallel lines.
Score full marks on 5-mark shortest-distance, intersection and foot-of-perpendicular/image questions and on 3-D case studies.
Solve common-perpendicular, image-of-a-point and multi-part triangle problems in 3-D.

The printable sheet

Three Dimensional Geometry knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Three Dimensional Geometry knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Linear Programming · Probability · All CBSE Class 12 Maths organisers