If `a + 2b + 3c= 0`, then `axxb + bxxc+c xxa` is equal to

Options

A. `2(axxb)`
B. `6(bxxc)`
C. `3(cxxa)`
D. 0

Answer

To solve the problem, we start with the equation given:

**Given:**
\[ a + 2b + 3c = 0 \]

We need to find the value of \( axxb + bxxc + cxxa \).

### Step 1: Take the cross product of the equation with \( b \)

We take the cross product of both sides of the equation with \( b \):
\[ (a + 2b + 3c) \times b = 0 \times b \]
This simplifies to:
\[ a \times b + 2b \times b + 3c \times b = 0 \]

### Step 2: Simplify the cross products

Since \( b \times b = 0 \) (the cross product of any vector with itself is zero), we can simplify the equation:
\[ a \times b + 0 + 3c \times b = 0 \]
This leads to:
\[ a \times b + 3c \times b = 0 \]
Rearranging gives us:
\[ a \times b = -3c \times b \quad \text{(Equation 1)} \]

### Step 3: Take the cross product of the equation with \( a \)

Now, we take the cross product of both sides of the original equation with \( a \):
\[ (a + 2b + 3c) \times a = 0 \times a \]
This simplifies to:
\[ a \times a + 2b \times a + 3c \times a = 0 \]

### Step 4: Simplify the cross products again

Since \( a \times a = 0 \), we have:
\[ 0 + 2b \times a + 3c \times a = 0 \]
This leads to:
\[ 2b \times a + 3c \times a = 0 \]
Rearranging gives us:
\[ 3c \times a = -2b \times a \quad \text{(Equation 2)} \]

### Step 5: Express \( c \times a \) in terms of \( b \times a \)

From Equation 2, we can express \( c \times a \):
\[ c \times a = -\frac{2}{3} b \times a \]

### Step 6: Substitute back into the expression \( axxb + bxxc + cxxa \)

Now we substitute the values we derived into the expression \( a \times b + b \times c + c \times a \):
Using Equation 1:
\[ a \times b + b \times c + c \times a = -3c \times b + b \times c + c \times a \]
Substituting \( c \times a \):
\[ = -3c \times b + b \times c - \frac{2}{3} b \times a \]

### Step 7: Combine the terms

Now we combine the terms:
\[ = -3c \times b + b \times c - \frac{2}{3} b \times a \]
Notice that \( b \times c = -c \times b \), so:
\[ = -3c \times b - c \times b - \frac{2}{3} b \times a \]
This simplifies to:
\[ = -4c \times b - \frac{2}{3} b \times a \]

### Final Result

Thus, the final expression simplifies to:
\[ = 6(b \times c) \]

### Conclusion

So, the answer is:
\[ a \times b + b \times c + c \times a = 6(b \times c) \]