Key Idea: There are two ways to multiply vectors. The dot product gives a number, used for angles and right angles. The cross product gives a vector, used for areas and for a direction perpendicular to two others.
Dot product
- v·w = v₁w₁ + v₂w₂ + v₃w₃.
- Angle: cos θ = (v·w)/(|v||w|). A negative dot product means an obtuse angle.
- Perpendicular ⟺ v·w = 0. Set it to 0 to find an unknown.
- For the angle at A in a triangle, use AB and AC, both pointing out of A.
| Use | Rule |
|---|---|
| Perpendicular direction | v×w is at right angles to both |
| Parallelogram area | |v×w| |
| Triangle area | ½|AB×AC| |
| Check | v·(v×w) = 0 |
Angle questions need the dot; area questions need the cross.
u = (k, 2, −1) and w = (3, k, 4) are perpendicular. Find k. 3k + 2k − 4 = 0, so k = 4/5.
A parallelogram panel has |p×q| = √275. What is its area? About 16.6 m². No halving for a parallelogram.
Does (1, 2, 3)·(2, 0, 1) give a vector? No. It gives the number 2 + 0 + 3 = 5.
Which vectors do you cross for a triangle's area? Two sides from the same corner, such as AB and AC.