If a, b, c are vectors such that [a b c] = 4 then `[a times b" " btimesc" "c timesa]=`

Options

A. 16
B. 64
C. 4
D. 8

Answer

To solve the problem, we need to find the value of the expression \([a \times b, b \times c, c \times a]\) given that the scalar triple product \([a, b, c] = 4\).

### Step-by-step Solution:

1. **Understanding Scalar Triple Product**:
The scalar triple product \([a, b, c]\) is defined as \(a \cdot (b \times c)\). It gives the volume of the parallelepiped formed by the vectors \(a\), \(b\), and \(c\).

2. **Given Information**:
We know that \([a, b, c] = 4\). This means that the volume of the parallelepiped formed by these vectors is 4.

3. **Expression to Evaluate**:
We need to evaluate the expression \([a \times b, b \times c, c \times a]\).

4. **Using Vector Triple Product Identity**:
We can use the vector triple product identity:
\[ x \times (y \times z) = (x \cdot z)y - (x \cdot y)z \]
We will apply this identity to find \(a \times (b \times c)\), \(b \times (c \times a)\), and \(c \times (a \times b)\).

5. **Calculating Each Term**:
- For \(a \times (b \times c)\):
\[ a \times (b \times c) = (a \cdot c)b - (a \cdot b)c \]
- For \(b \times (c \times a)\):
\[ b \times (c \times a) = (b \cdot a)c - (b \cdot c)a \]
- For \(c \times (a \times b)\):
\[ c \times (a \times b) = (c \cdot b)a - (c \cdot a)b \]

6. **Putting It All Together**:
Now we can express \([a \times b, b \times c, c \times a]\) using the results from the previous step:
\[ [a \times b, b \times c, c \times a] = [(a \cdot c)b - (a \cdot b)c, (b \cdot a)c - (b \cdot c)a, (c \cdot b)a - (c \cdot a)b] \]

7. **Final Calculation**:
The scalar triple product of three vectors in the form \([x, y, z]\) can be computed using the determinant:
\[ [x, y, z] = x \cdot (y \times z) \]
Thus, substituting the expressions we derived, we can find the final result.

8. **Conclusion**:
Using the properties of determinants and the scalar triple product, we can conclude that:
\[ [a \times b, b \times c, c \times a] = [a, b, c]^2 = 4^2 = 16 \]

### Final Answer:
\[ [a \times b, b \times c, c \times a] = 16 \]