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The triple vector product

, which can also be written in the form

, is one way of multiplying the three vectors

. The result is a vector lying in the same plane as

and

. Here the vector triple product is shown in red, and the vector

is also shown in magenta. The blue plane is the plane determined by

and

, to which the magenta vector is perpendicular. You can choose to see the unit vector in the direction of

with "direction only".

Snapshot 1: both magnitude and direction of the triple vector product

Snapshot 2:

and

are parallel

Snapshot 3:

and

are perpendicular

The triple vector product

for non-zero vectors

vanishes when

is parallel to

Reference

[1] D. Fleisch, A Student's Guide to Vectors and Tensors, New York: Cambridge University Press, 2012.

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Triple Vector Product