what is vector triple product?

Answer 1

volume of the Parallelopiped the 3 vectors span

(A x B) . C

read: vector product of A and B, which is a vector, multiplied scalarly by C that produces a number (volume)

Answer 2

The vector triple product
If A, B and C are three vectors, then we can combine them in this way:

Ax(BxC)

to get a vector result, which is known as the vector triple product.

As you know, the cross product is calculated using a determinant and we can extend this to Ax(BxC) to get a rather more complicated determinant than the scalar triple product gave us, again involving all three vectors.

However there is a much simpler way to evaluate a vector triple product, because it can be shown that this is true:

Ax(BxC)=(A.C)B-(A.B)C

and

(AxB)xC=(C.A)B-(C.B)A.

So we can evaluate either of those right-hand sides instead, which do not involve any determinants. However you do need to remember them!

Here's an example of a vector triple product. If the vectors A, B and C are as given below, what is their vector triple product, Ax(BxC)?

A=2i+3j-4k, B=i-2j+2k, C=3i-3j-k.

You have a go first, then check your answer.

Here's another one for you to practise on. Find the vector triple product, Ax(BxC), if the vectors A, B and C are as given below:

A=i-3j-2k, B=2i+2j+k, C=i-4j+3k.

Here are the three vectors again:

A=i-3j-2k, B=2i+2j+k, C=i-4j+3k.

Here's the vector triple product again:

Ax(BxC)=(A.C)B-(A.B)C

First then we need to know A.C:

A.C=1x1-3x(-4)-2x3=7

Second we need to know A.B:

A.B=1x2-3x2-2x1=-6

Putting these into the formula, then, we get:

Ax(BxC)=7(2i+2j+k) -(-6)(i-4j+3k)

so

Ax(BxC)=i(14+6)+j(14-24)+k(7+18)= 20i-10j+25k

Answer 3

http://em-ntserver.unl.edu/Math/mathweb/vectors/vectors.html#vec9


Doug