What is the vector triple product used for?

To examine the relationship between vectors, particularly in three dimensions, the vector triple product is utilised. Thus, it helps in the simplification of the majority of issues involving torque, angular momentum, and force systems.

How does the vector triple product differ from the scalar triple product?

A scalar is produced by the scalar triple product, whereas a vector is produced by the vector triple product. Although the inputs for these procedures are all three vectors, their applications in geometry, mechanics, and vector algebra vary.

Why is the order important in the vector triple product?

The final result of the vector triple product depends on the order. It is necessary to adhere to proper computing rules since altering the grouping or sequence of operations may result in entirely different results.

Where is the vector triple product used?

The vector triple product has applications in physics in torque and angular momentum, in geometry in checking the alignment of vectors, and in engineering in computations for structural mechanics and the electromagnetic field.

Vector Triple Product

Of all the concepts under vector algebra, the vector triple product is one that involves the cross-product of three vectors. A fact that may be noticed and used when calculating the vector triple product is that the cross product of a specific vector and the result of the cross product of the other two vectors must be calculated. In applications including mechanics, electromagnetism, and geometry, it is a vector operator which helps in the expression of vector connections and streamlines challenging issues. It is widely used in engineering and physics.

1.0Vector Triple Product definition

The vector triple product involves three vectors - A, B and C. It comes from taking the cross-product of A with the cross-product of B and C. The vector triple product is expressed as:

A × (B × C)

Vector Triple Product Formula is:

A × ( B × C) = (A⋅ C) B − (A⋅ B) C

The vector triple product follows the BAC-CAB Rule. This rule aids weight calculations because it enables numerous vector operations to be done, such as numerous scalar and vector multiplication.

2.0Proof of Vector Triple Product

To prove the BAC-CAB Rule, let:

B × C = D = (D_x , D_y, D_z)

Now, compute A × D using the determinant form of the cross product:

A x D = [ i j k ]

[A_xA_yA_z]

[D_xD_yD_z]

Expanding and substituting D in terms of B and C, the terms rearrange into:

A × (B × C) = (A ⋅ C) B− (A ⋅ B) C

This proves the vector triple product formula.

3.0Applications of the Vector Triple Product

Physics: Used in torque computations, moment of inertia, force system and many other fields.
Geometry: Assist in finding the correlation of vectors in the 3-dimensional space.
Engineering: Technology used in structural mechanics and fluid dynamics.

4.0Properties of the Vector Triple Product

The vector triple product has unique properties that distinguish it from other vector operations:

Vector Nature: The result of a vector triple product is always a vector quantity, making it different from the scalar triple product, which yields a scalar.
Unit Vector: In a vector triple product, a unit vector that lies in the same plane as A and B and is perpendicular to C can be identified. This unit vector is expressed as:

± (A × B) × C / ∣ (A × B) × C ∣

Distinct from Cross Products: It is important to note that the order and grouping in the vector triple product matter. Specifically:

A × (B × C) = (A × B) × C

This distinction is crucial in computations.

Non-Coplanar Vectors: If A, B and C are non-coplanar vectors (not lying in the same plane), then the vectors A × B, B × C and C × A are also non-coplanar, preserving their spatial relationships.

These properties highlight the versatility and utility of the vector triple product in advanced vector operations.

5.0Difference between Scalar Triple Product and Vector Triple Product

Feature	Scalar Triple Product	Vector triple product
Definition	Dot product of a vector A with the cross product of vectors B and C. Given by A.(BC)	it is the cross product of two vectors and then the cross product of the result with a third one. Given by A(BC)
Result	Scalar (a number)	Vector
Geometrical interpretation	Volume of parallelepiped	Vector in the plane of B and C, related to projections
Commutativity	commutative	Non-commutative
Antisymmetry	Yes, changing any two vectors changes the sign	No (order matters in cross-products)
Zero condition	Zero if vectors are coplanar	Depends on the specific vectors
Usage	Volume, scalar quantities	Used in various vector calculations like torque, angular momentum

6.0Vector Triple Product example

Example 1: Simplify a Vector Triple Product

Simplify A × ( B × C) for A = i + 2j − k, B = 3i − j + 2k and C= 2i + j − k

Solution:

Using the BAC - CAB Rule:

A ×( B × C) = (A · C) B − (A ⋅ B)C

Step 1: Compute A⋅ C:

A ⋅ C = (1) (2) + (2) (1) + (−1) (−1) = 2 + 2 + 1 = 5

Step 2: Compute A ⋅ B:

A ⋅ B = (1) (3) + (2) (−1) + (−1) (2) = 3−2−2 = −1

Step 3: Substitute values:

A × (B × C) = (5) B − (−1)C

= 5 (3i − j + 2k) + C

= (15i − 5j + 10k) + (2i + j − k)

= 17i − 4j + 9k

Answer: A × (B × C) = 17i − 4j + 9k.

Example 2: If A × B = C, B × C = A and A, B, and C be moduli of the vectors A, B, and C, respectively, then find the values of A & B.

Solution:

Magnitudes of both sides

For first equation

AB=C ABsin=C …….(1)

For the second equation

BC=A BCsin=A …….(2)

Substituting the value of C from Equation 1 into Equation 2

B(ABsin)sin=A

AB2sinsin=A B2sinsin=1

∴ A is perpendicular to both B & C, and C is perpendicular to both A & B.

Therefore, A, B, & C are mutually perpendicular. Then sin=sin=1(as ==90)

B² = 1 B = 1

From equation 1

A1sin90=C

A = C

The vector triple product is such an important tool in vector algebra, letting them simplify complex vector relationships in a three-dimensional space. Its proof based on the BAC-CAB rule presents it as elegant and practical. Mastering concepts like the example of the vector triple product lets students and professionals resolve their problems in physics, geometry, and engineering problems.

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