“Dot gives a number, cross gives a vector”
Telling the products apart
The dot product answers how much two vectors agree and is a scalar. The cross product answers what is perpendicular to both and is a vector.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Vectors and Three Dimensional Geometry formulas
Vectors and 3D geometry are one chapter twice: every line and plane result is a vector identity written out. Learn the two products properly and the geometry becomes substitution.
20 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
Vector algebra
| What | Formula | Symbols and notes |
|---|---|---|
| Magnitude | |a| = √(a₁² + a₂² + a₃²) | for a = a₁î + a₂ĵ + a₃k̂ |
| Unit vector | â = a/|a| | magnitude one, same direction |
| Dot product | a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃ | a scalar |
| Perpendicular test | a·b = 0 | for non-zero vectors |
| Cross product | |a × b| = |a||b| sinθ | a vector perpendicular to both |
| Parallel test | a × b = 0 | the sine of zero is zero |
| Area of a triangle | ½|a × b| | and a parallelogram is |a × b| |
| Scalar triple product | [a b c] = a·(b × c) | volume of the parallelepiped; zero if coplanar |
| Projection | projection of a on b = (a·b)/|b| | a scalar length |
Lines in space
| What | Formula | Symbols and notes |
|---|---|---|
| Vector form | r = a + λb | a is a point on the line, b its direction |
| Cartesian form | (x − x₁)/a = (y − y₁)/b = (z − z₁)/c | a, b, c are direction ratios |
| Direction cosines | l = a/√(a² + b² + c²), and l² + m² + n² = 1 | cosines of the angles with the axes |
| Angle between two lines | cosθ = |b₁·b₂|/(|b₁||b₂|) | from the direction vectors |
| Shortest distance, skew lines | |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂| | zero when the lines intersect |
Planes
| What | Formula | Symbols and notes |
|---|---|---|
| Vector form | r·n̂ = d | n̂ is the unit normal, d the distance from the origin |
| Cartesian form | ax + by + cz = d | (a, b, c) are the direction ratios of the normal |
| Through three points | use the scalar triple product to eliminate | or take the cross product of two edge vectors |
| Angle between two planes | cosθ = |n₁·n₂|/(|n₁||n₂|) | the angle between their normals |
| Angle between a line and a plane | sinθ = |b·n|/(|b||n|) | note the sine, not the cosine |
| Distance from a point | |ax₁ + by₁ + cz₁ − d|/√(a² + b² + c²) | the three-dimensional version of the line formula |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Zero dot perpendicular, zero cross parallel”
The two tests
A zero dot product means at right angles; a zero cross product means in line. cos 90° = 0 and sin 0° = 0 explain both.
“Line and plane uses sine”
The one angle formula that differs
Line to line and plane to plane use cosine. Line to plane uses sine, because the angle is measured to the surface while the normal sticks out of it.
Where marks go
The mistakes that cost the most.
- Using cosine for the angle between a line and a plane. That one takes the sine.
- Treating the cross product as commutative. a × b = −(b × a).
- Reading the plane's coefficients as a point on it. They are the direction ratios of the normal.
- Forgetting that the scalar triple product being zero means the three vectors are coplanar.
- Mixing direction ratios with direction cosines. The cosines are the ratios divided by the magnitude.
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