The following formula is used to calculate the vector triple product:
\[ \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B}) \]
Variables:
\(\mathbf{A}, \mathbf{B}, \mathbf{C}\) are vectors
\(\times\) is the cross product operator
\(\cdot\) is the dot product operator
The Vector Triple Product is a mathematical operation in vector algebra that involves the cross-product of three vectors. It is expressed as \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C})\), where \(\mathbf{A}, \mathbf{B},\) and \(\mathbf{C}\) are three vectors. The result of this operation is a new vector that is perpendicular to the plane defined by the original three vectors. The magnitude and direction of this new vector are determined by the right-hand rule. The Vector Triple Product has the property of not being associative, meaning that \((\mathbf{A} \times \mathbf{B}) \times \mathbf{C}\) does not necessarily equal \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C})\).
Let's say we have three vectors:
\(\mathbf{A} = (1, 2, 3)\)
\(\mathbf{B} = (4, 5, 6)\)
\(\mathbf{C} = (7, 8, 9)\)
First, we calculate the cross product \(\mathbf{B} \times \mathbf{C}\):
\[ \mathbf{B} \times \mathbf{C} = (5*9 - 6*8, 6*7 - 4*9, 4*8 - 5*7) = (-3, 6, -3) \]
Next, we calculate the dot products:
\[ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 1*(-3) + 2*6 + 3*(-3) = 0 \]
\[ \mathbf{A} \cdot \mathbf{B} = 1*4 + 2*5 + 3*6 = 32 \]
\[ \mathbf{A} \cdot \mathbf{C} = 1*7 + 2*8 + 3*9 = 50 \]
Now, calculate the vectors for the final formula:
\[ \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) = 50 \times (4, 5, 6) = (200, 250, 300) \]
\[ \mathbf{C}(\mathbf{A} \cdot \mathbf{B}) = 32 \times (7, 8, 9) = (224, 256, 288) \]
Finally, we subtract the two vectors:
\[ \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (200, 250, 300) - (224, 256, 288) = (-24, -6, 12) \]
Therefore, the vector triple product \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (-24, -6, 12)\).