Interactive vector geometry: a rhombus with side vectors u and v and its diagonals u + v and v − u, and a triangle showing medians, altitudes, perpendicular bisectors and angle bisectors, with the angle at A as the control

Perpendicular is not an angle you measure. It is a zero

A vector proof never measures anything. Write each point as a position vector and the geometry turns into algebra: equal sides become equal magnitudes, a midpoint becomes an average, and a right angle becomes a scalar product that equals zero. Change the angle at A and watch the two diagonal lengths and the number AC · BD; then switch the picture to a triangle and watch where its medians, altitudes and bisectors meet.

Vector proof on screen Rhombus • scalar product
Change the angle at A. Watch the two diagonal lengths and the number AC · BD.
Which proof the picture shows
AC · BD = |v|² − |u|² = 0.00 • |u × v| = sin θ = 0.87
Angle at A60°
30°60°90°120°150°
Diagonal AC
1.73
Diagonal BD
1.00
AC · BD
0.00
Area |u × v|
0.87
Read the picture as a proof
Know This
In a vector proof, equal lengths become equal magnitudes, a right angle becomes a scalar product of zero, and a midpoint becomes the average of two position vectors.