1 Vector Triple Product

Combining three vectors into a product is called a triple product. The vector triple product is the vector product of two vectors of which one is itself a vector product. Such as

The vector A × is perpendicular to A and to . But is perpendicular to the plane of B and C. Hence A ×, being perpendicular to must lie in the plane of B and C and thus take the form

where x an y are two scalars. In like manner also the vector × C, being perpendicular to must lie in the plane of A and B. Hence it will be of the form

where m and n are two scalars. From this it is evident that in general

The parentheses therefore cannot be removed or interchanged. It is essential to know which cross product is formed first and which second. This product is termed the vector triple product in contrast to the scalar triple product.

1.1 Geometric Interpretation

The vector triple product may be used to express that component of a vector B which is perpendicular to a given vector A. This geometric use of the product is valuable not only in itself but for the Light it sheds upon the properties of the product. Let A (below figure) be a given vector and B another vector whose components parallel and perpendicular to A are to be found. Let the components pf B parallel and perpendicular to A be B′ and B′′ respectively. Draw A and B from a common origin. The product A × B is perpendicular to the plane of A and B. The product A × lies in the plane of A and B. It is furthermore perpendicular to A. Hence it is collinear with B′′. An examination of the figure will show that the direction of A × is opposite to that of B′′.

Hence

where c is some scalar constant.

Now

where b′′ is the unit vector in the direction of B′′.

But

Hence

Therefore

(1)

The component of B perpendicular to A has been expressed in terms of the vector triple product of A, A, and B. The component B′ parallel to A is found using the dot product as a projection and leads to

(2)

Hence

(3)

Next, consider the product when two of the vectors are the same. By equation (3)

(4)

Or

(5)

This proves the formula in case two vectors are the same.

1.2 Property I .

The vector triple product A × may be expressed as the sum of two terms as

To prove this, express A in terms of the three non-coplanar vectors B, C, and .

(6)

where a, b, c are scalar constants. Then

(7)

The vector product of any vector by itself is zero. Hence

so

(8)

Using the relationship developed under the geometric interpretation

with different vectors yields

Hence

(9)

But from (1)

using a property from the scalar triple product

we get

Substituting these values into (4)

(10)

Finally, noting that the dot product gives a scalar, we can use the Vector Identity

to get

(11)

The relation is therefore proved for any three vectors A, B and C and is often referred to as the BACK CAB rule.

1.3 Property II.

From the three letters A, B, C by different arrangements, four allied products in each of which B and C are included in parentheses may be formed. These are

As a vector product changes its sign whenever the order of two factors is interchanged, the above products evidently satisfy the equations

The expansion for a vector triple product in which the parenthesis comes first may therefore be obtained directly from that already found when the parenthesis comes last.

The formulas then become

These reduction formulas are of such constant occurrence and great importance that they should be committed to memory.

1.4 References

[1] Wilson, E. ”Vector Analysis.” Yale University Press, New Haven, 1913.

This entry is a derivative of the Public domain work [1].