Three Dimensional Geometry
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Direction Cosines and Ratios
Part of: Directions and Lines
A line steers through space by its angles with the axes; cosines of those angles are direction cosines squaring to one. Any proportional triple works as direction ratios, two points give differences over distance, and proportional joins mark collinear points.
Why it matters. Directions steer every line, angle and distance that follows.
Formulas
- Direction cosines square-sum to one
- Ratios scale into cosines by dividing length
- Proportional joins mark collinear points
Key words
- Direction cosines
- : Cosines of angles with the three axes.
- Direction ratios
- : Any triple proportional to direction cosines.
- Directed line
- : A line with one chosen arrow direction.
- Collinear points
- : Points sharing one straight line.
Try it in two minutes
- Read angles and cosine each one.
- Scale ratios down to cosines.
- Compare joins for proportionality.
Try at home. Find DCs for DRs 1, 2, 2 and verify the square sum.
Equations of a Line
Part of: Directions and Lines
A line needs a point plus a parallel direction: vector form adds scaled direction to the point, and Cartesian form equates the three coordinate ratios. Two points give differences as direction, and a point satisfies the line when all three ratios agree.
Why it matters. Line equations carry flights, beams and edges through space.
Formulas
- Vector form is anchor plus λ times direction
- Cartesian form equates three ratios
- Points on lines give equal ratios
Key words
- Vector equation
- : Point plus scaled direction vector.
- Cartesian equation
- : Three equal coordinate ratios.
- Parameter
- : The scalar sliding points along the line.
- Two-point form
- : Ratios built from point differences.
Try it in two minutes
- Anchor the given point first.
- Attach the direction with a parameter.
- Equate the three ratios for Cartesian form.
Try at home. Write the line through (0, 0, 0) with DRs 1, −1, 2 in both forms.
Angle Between Two Lines
Part of: Angles and Shortest Distance
The angle between lines is the angle between their direction vectors, with cosine as dot over length product. Zero dot marks perpendicular pairs, proportional ratios mark parallel pairs, and direction cosines drop the denominators to one.
Why it matters. Crossing beams, flight paths and edges meet at computable angles.
Formulas
- Cosine is dot over length product
- Zero dot marks perpendicular lines
- Proportional ratios mark parallel lines
Key words
- Acute angle rule
- : Line angles are read acute via modulus.
- Perpendicular lines
- : Lines with zero dot of directions.
- Parallel lines
- : Lines with proportional direction ratios.
- Direction dot
- : The numerator deciding the angle.
Try it in two minutes
- Read direction ratios off both lines.
- Dot them over the length product.
- Test zero for square corners, ratios for parallels.
Try at home. Find the angle between DRs (1, 0, 0) and (1, 1, 1).
Shortest Distance
Part of: Angles and Shortest Distance
Skew lines never meet and refuse parallels, floating non-coplanar. Their shortest join stands perpendicular to both, found by projecting the connecting segment onto the cross direction. Parallel pairs sit coplanar with a perpendicular gap, and intersecting pairs touch at zero distance.
Why it matters. Wires, rails and flight corridors keep safe gaps through these measures.
Formulas
- Skew distance projects join onto cross direction
- Parallel gap is perpendicular distance
- Crossing lines sit at zero distance
Key words
- Skew lines
- : Non-coplanar lines that never meet nor parallel.
- Shortest distance
- : The smallest join between two lines.
- Cross direction
- : The perpendicular to both line directions.
- Coplanar lines
- : Lines sharing one plane.
Try it in two minutes
- Test parallel ratios and meeting points first.
- Name the pair skew, parallel or crossing.
- Project the join onto the cross direction.
Try at home. Classify two pencils on a desk as skew, parallel or crossing.
What your child will practise
- Angle Between Two Lines: Candy Puzzle (Clue Puzzle)
- Angle Between Two Lines: Ocean Quick Fire (Quick Fire)
- Direction Cosines and Ratios: Desert Puzzle (Clue Puzzle)
- Equations of a Line: Candy Quick Fire (Quick Fire)
- Shortest Distance: Desert Quest (Flip and Match)
- Quiz: Angle Between Two Lines (8 questions)
- Quiz: Direction Cosines and Ratios (8 questions)
- Quiz: Equations of a Line (8 questions)
- Quiz: Shortest Distance (8 questions)