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# Vector Triple Product

The cross product between three vectors is dealt with in vector triple product, a branch of vector algebra. The cross-product of a vector with the cross-products of the other two vectors yields the result of the vector triple product.

There is a branch of vector algebra known as Vector Triple Product. It is a study of the cross-product of 3 vectors in a vector triple product.

The quantity of a vector triple product may be computed by calculating the cross-product of a vector with the cross products of the other two vectors. As a result, a vector quantity is generated. The BAC – CAB identification name may be acquired from the result after the vector triple product has been simplified.

## Vector Triple Product Definition

The determinant of a matrix is what the “vector triple product” is. In particular, if you consider a matrix’s columns to represent your three vectors, the matrix’s determinant equals the triple product.

### What exactly is a Vector Triple Product?

Let a, b, and c be three vectors.

The cross product of vector a with the cross products of vectors b and c is known as their Vector triple product.

Mathematically, it can be represented as a × (b × c)

The vectors b and c are coplanar with the triple product. In addition, the triple product lies perpendicular to a.

The mathematical form of this would be a × (b × c) =xb +yc

## The formula of Vector Triple Product

The formula for vector triple product is:

Where a × (b × c) ≠ (a × b) ×c

## Properties of Vector Triple Product

1.   A vector triple product yields a vector quantity as a result.
2.   a⃗ × (b⃗ × c⃗) ≠ (a⃗ × b⃗) ×c⃗
3.   Vector r=a×(b×c) is coplanar to b and c and perpendicular to a.
4.   Only if the vector outside the bracket is on the leftmost side, does the formula r=a1+λb hold true. If it isn’t, we use the principles of cross-product to shift to the left and then use the same procedure.

## Example

1. Suppose vectors a, b and c are coplanar. Prove that a x b, a x c, and b x c are also coplanar.

ANS: As they are coplanar, we can write them as

[a x b x c] = 0

By squaring both sides, we get:

[a x b x c]2 = 0

[(a⃗ × b⃗) (b⃗ × c⃗) (c⃗ × a⃗)] =0

Therefore, the products are also coplanar.

## Difference between Scalar Triple Product and Vector Triple Product

The dot product of a vector with the cross product of two different vectors[3] [SR4]  is called the scalar triple product. For example, if a, b and c are three vectors, the scalar triple product is a. (b x c). The box product and mixed product are other names for it. The volume of a parallelepiped is calculated using the scalar triple product, where the three vectors indicate the parallelepiped’s neighboring sides.

The cross product of vector a with the cross products of vectors b and c is known as their Vector triple product. The vectors b and c are coplanar with the triple product. In addition, the triple product lies perpendicular to a.

## Conclusion

The quantity of a vector triple product may be computed by cross-producting a vector with the cross product of the other two vectors. As a result of this cross-product, a vector quantity is generated.

The quantity of a vector triple product may be computed by calculating the cross-product of a vector with the cross products of the other two vectors. As a result, a vector quantity is generated. The BAC – CAB identification name may be acquired from the result after the vector triple product has been simplified.

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