Vectors and 3D Geometry (Class 12 Maths): Key Formulas and Board-Style Questions
Dot and cross products, the angle between lines and planes, and the shortest distance between skew lines — the highest-yield block in the paper.
The two products, and what each is for
| Dot product | Cross product | |
|---|---|---|
| Result | A scalar | A vector |
| Formula | a · b = |a||b| cos θ | a × b = |a||b| sin θ n̂ |
| Zero when | Vectors are perpendicular | Vectors are parallel |
| Used for | Angles, projections, work | Areas, normals, moments |
Formulas worth knowing cold
d = | ( a2 − a1 ) · ( b1 × b2 ) | / | b1 × b2 |
sin θ = | b · n | / ( |b| |n| )
The trap in line-and-plane angles
For two lines, or two planes, the angle uses cosine. For a line and a plane, it uses sine. The reason is geometric: the angle you are asked for is measured from the line to the plane, but the vector you have is the plane's normal, which is perpendicular to the plane — so the two angles are complementary.
- Line and line → cos θ = | b1 · b2 | / ( |b1| |b2| )
- Plane and plane → cos θ = | n1 · n2 | / ( |n1| |n2| )
- Line and plane → sin θ = | b · n | / ( |b| |n| )
FAQs
When do I use sine instead of cosine for an angle?
Sine is used only for the angle between a line and a plane, because the plane is represented by its normal. Line-to-line and plane-to-plane both use cosine.
How do I know if two lines are skew?
They are skew if they are neither parallel nor intersecting. If the shortest-distance formula gives a non-zero value, the lines are skew.
What does the cross product give geometrically?
A vector perpendicular to both inputs, with magnitude equal to the area of the parallelogram they span.
Is the dot product commutative?
Yes, a · b = b · a. The cross product is not — a × b = −(b × a).
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