Summary
- The vector product a × b gives a new vector perpendicular to both a and b, with magnitude ab sin θ.
- The vector product has a neat component formula using i, j, k or a determinant.
- a × b gives you areas (triangles, parallelograms) almost for free.
- The triple scalar product a · (b × c) gives the (signed) volume of a parallelepiped; a sixth of it gives the volume of a tetrahedron.
- Lines in 3-D can be written three ways: vector form r = a + λb, Cartesian form (x−l)/α = (y−m)/β = (z−n)/γ, and vector product form r × b = c.
- Planes can be written three ways too: scalar product form r·n = d, Cartesian form ax+by+cz=d, and vector form r = a + λb + μc.
- There's a whole toolkit of distance formulas: point–to–plane, point–to–line, between parallel planes, between skew lines. They all boil down to projecting one vector onto a unit vector.
- Reflections in a plane use the point-to-plane method twice (find the foot of the perpendicular, then double it).
- The angle between two lines, or two planes, or a line and a plane, always comes from a dot product of direction/normal vectors — you just have to know which pair to dot, and whether to subtract from 90°.
The Vector (Cross) Product
You already know the scalar product a·b = ab cos θ from earlier work — it takes two vectors and spits out a . The vector product does something different: it takes two vectors and spits out a .
Think of it like this: if a and b are two edges of a flat sheet of card lying on a table, a × b points straight up out of the table (or straight down, depending on which way you rotate from a to b). Its tells you how "spread apart" a and b are (maximum when they're perpendicular, zero when they're parallel).
🧭 Why "corkscrew"?
Picture i × j. Turning from i to j is a quarter-turn anticlockwise when viewed from the +z direction — exactly how a normal (right-handed) screw moves forward as you turn it that way. So i × j = k. This gives you the three key results: i × j = k, j × k = i, k × i = j — and reversing any pair flips the sign.
Key properties
- Anticommutative: b × a = −(a × b). Swapping the order of a cross product flips the direction of n̂ (the corkscrew turns the other way), so the whole vector negates.
- a × a = 0, since θ = 0 so sin θ = 0.
- a × b = 0 ⟺ a is parallel to b (or one of them is the zero vector) — because sin θ = 0 only when θ = 0 or π.
- Distributive over addition: a × (b + c) = a × b + a × c.
- a × b is always perpendicular to both a and b — straight from the definition.