Vector Triple Product Calculation Explained

This problem requires us to find the value of the scalar triple product \(\left [\vec a\ \vec b\ \vec c \right]\) when we are given the scalar triple product of cross products, specifically \(\left [\vec a \times \vec b,\ \vec b \times \vec c,\ \vec c \times \vec a \right]\) = 64. To solve this, we need to recall a fundamental identity in vector algebra involving the scalar triple product.

Understanding Scalar Triple Product

The scalar triple product of three vectors \(\vec a\), \(\vec b\), and \(\vec c\) is denoted as \(\left [\vec a\ \vec b\ \vec c \right]\) and is defined by the expression \(\vec a \cdot (\vec b \times \vec c)\). Geometrically, the absolute value of the scalar triple product represents the volume of the parallelepiped formed by the three vectors \(\vec a\), \(\vec b\), and \(\vec c\). It is a scalar quantity, meaning it has magnitude but no direction.

Key Vector Identity for Cross Products

A crucial vector identity connects the scalar triple product of cross products with the scalar triple product of the original vectors. This identity is:

$$\left [\vec a \times \vec b,\ \vec b \times \vec c,\ \vec c \times \vec a \right] = \left [\vec a\ \vec b\ \vec c \right]^2$$

This identity is very useful for problems like this one, where we need to relate the scalar triple product of derived vectors (cross products) to the scalar triple product of the original vectors.

Step-by-Step Scalar Triple Product Solution

Let's use the given information and the identity to find the value of \(\left [\vec a\ \vec b\ \vec c \right]\).

• Given Information: We are given that \(\left [\vec a \times \vec b,\ \vec b \times \vec c,\ \vec c \times \vec a \right]\) = 64.
• Applying the Identity: We know the vector identity is \(\left [\vec a \times \vec b,\ \vec b \times \vec c,\ \vec c \times \vec a \right] = \left [\vec a\ \vec b\ \vec c \right]^2\).
• Substitution: Substitute the given value into the identity:

  $$\left [\vec a\ \vec b\ \vec c \right]^2 = 64$$

• Solving for \(\left [\vec a\ \vec b\ \vec c \right]\): To find \(\left [\vec a\ \vec b\ \vec c \right]\), we take the square root of both sides:

  $$\left [\vec a\ \vec b\ \vec c \right] = \pm \sqrt{64}$$

  $$\left [\vec a\ \vec b\ \vec c \right] = \pm 8$$

• Considering the Result: The scalar triple product can be positive or negative, depending on the orientation of the vectors (whether they form a right-handed or left-handed system). However, when typical multiple-choice questions provide only positive options, the positive root is usually the intended answer. In this case, the options include 8, which is the positive root.

Concluding the Scalar Triple Product Result

Based on our calculations and considering the available options, the value of \(\left [\vec a\ \vec b\ \vec c \right]\) is 8.

The final answer is 8.