Triple Vector Product

direction only

a

1

-1.58

a

2

0

a

3

3

b

1

2

b

2

-5.

b

3

2

c

1

2

c

2

2.28

c

3

0

The triple vector product

a×(b×c)

, which can also be written in the form

(a·c)b-(a·b)c

, is one way of multiplying the three vectors

a=(

a

1

,

a

2

,

a

3

)

,

b=(

b

1

,

b

2

,

b

3

)

,

c=(

c

1

,

c

2

,

c

3

)

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

b

and

c

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

b×c

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

b

and

c

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

a×(b×c)

with "direction only".

Details

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

Snapshot 2:

a

and

b

are parallel

Snapshot 3:

a

and

b

are perpendicular

The triple vector product

a×(b×c)

for non-zero vectors

a

,

b

,

c

vanishes when

b

is parallel to

c

or

c

is parallel to

b×c

.

References

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

External Links

Vector Triple Product (Wolfram MathWorld)

Permanent Citation

Roberta Grech

"Triple Vector Product"
http://demonstrations.wolfram.com/TripleVectorProduct/
Wolfram Demonstrations Project
Published: August 20, 2013