Dot and cross products in 3D

Use the dot product to find angles and perpendicular vectors, and the cross product to find areas and normals. About 8 minutes, A-level Further Maths.

What you will do

1. Multiply and add. Two vectors u (red) and v (green) start at the origin. Drag the view to orbit and see them from any side. The dot product multiplies matching components and adds: it turns two vectors into one number.
2. The angle hidden in a dot product. The dot product also equals | u|| v|, so it tells you the angle between the vectors. Here are (1,0,0) and (1,1,0).
3. Make them perpendicular. If u× v = 0 the vectors are at right angles. Change the sliders until the dot product is exactly 0 and the panel says they are perpendicular.
4. The cross product. The cross product u× v is a new vector at right angles to both. Its length is the area of the shaded parallelogram.
5. No area. If two vectors point the same way, the parallelogram collapses: area 0, and the cross product is the zero vector. Make | u× v| smaller than 0.1.
6. Your turn. Everything is unlocked. Orbit around, change the vectors, show the projection of u onto v, and check that u × v is perpendicular to both.