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
- A vector triple product yields a vector quantity as a result.
- a⃗ × (b⃗ × c⃗) ≠ (a⃗ × b⃗) ×c⃗
- Vector r=a×(b×c) is coplanar to b and c and perpendicular to a.
- 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
- 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.