CBSE 12 Math · U4
Vectors and Three-Dimensional Geometry
Vector operations, scalar and vector products, equations of lines in space, angles, skew lines, and shortest distance.
Test yourself
A focused, distraction-free run through 35 multiple-choice questions. Hints if you need them; your score at the end.
Difficulty guide
The label describes the problem, not your ability.
1/5 - Foundational
Warm-up setup; checks one basic idea.
2/5 - Routine exam skill
Normal syllabus skill; usually one main rule or interpretation.
3/5 - Standard multi-step
Exam-style problem with multiple steps or a common trap.
4/5 - Hard exam skill
High-end exam problem; combines methods carefully.
5/5 - Challenge
Stretch problem beyond normal exam pressure.
Topic 4.1
Vector Basics, Components, Magnitude, and Direction Cosines
Position vectors, displacement vectors, unit vectors, direction ratios, and direction cosines
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Topic 4.2
Vector Operations and Section Formula
Addition, scalar multiplication, parallel vectors, midpoint, and section formula
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Topic 4.3
Scalar Product, Projections, and Angles
Dot product, angle between vectors, scalar projection, vector projection, and work
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Topic 4.4
Vector Product and Area
Cross product, perpendicular vectors, parallelogram area, triangle area, and vector area
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Topic 4.5
Lines in Three-Dimensional Geometry
Direction ratios, direction cosines, vector equation, Cartesian equation, and parametric form of lines
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Topic 4.6
Angles, Skew Lines, and Shortest Distance
Angle between lines, parallel and skew lines, and shortest distance between lines
Topic 4.7
Line Forms and Conversion in Three-Dimensional Geometry
Lines through two points, vector-Cartesian conversion, parametric form, and point membership
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