Vector Triple Product Explained Visually | BAC–CAB Identity in 100 Seconds

Added: 2026-06-03

[00:00:00]The Cross product is not associative. That means a cross b cross c must be read with brackets. The famous identity is a cross b cross c equals B A C minus C A B.

[00:00:13]The memory trick is B A C minus C A B. For a cross b cross c, the answer keeps b and c as vectors. B gets the dot product a dot c. C gets the dot product a dot b, with a minus sign.

[00:00:31]Now derive it in index notation. Here epsilon i j k denotes the same alternating symbol often written as e i j k. The component of a cross b cross c is epsilon i j k a j times epsilon k l m b l c m. The repeated index k is contracted.

[00:00:54]The Kronecker deltas substitute indices. The first filter gives b i times a j c j. The second filter gives c i times a j b j. That is exactly b times a dot c minus c times a dot b.

[00:01:11]Geometrically, the result is not an arbitrary vector. It lies in the plane spanned by b and c. The dot products a dot c and a dot b decide how much of b and c survive.

[00:01:26]Remember the order: B A C minus C A B. The vector triple product is not a trick. It is epsilon delta contraction.