The condition for equations `vecrxxveca = vecb and vecr xx vecc = vecd` to be consistent is

Options

A. `vecb.vecc=veca.vecd`
B. `veca.vecb=vecc.vecd`
C. `vecb.vecc+veca.vecd=0`
D. `veca.vecb+vecc.vecd=0`

Explanation

To determine the condition for the equations \( \vec{r} \times \vec{a} = \vec{b} \) and \( \vec{r} \times \vec{c} = \vec{d} \) to be consistent, we can follow these steps:

### Step 1: Cross both sides of the first equation with \( \vec{d} \)
We start with the first equation:
\[ \vec{r} \times \vec{a} = \vec{b} \]
Now, we cross both sides with \( \vec{d} \):
\[ \vec{d} \times (\vec{r} \times \vec{a}) = \vec{d} \times \vec{b} \]

### Step 2: Apply the vector triple product identity
Using the vector triple product identity, we can rewrite the left-hand side:
\[ \vec{d} \times (\vec{r} \times \vec{a}) = (\vec{d} \cdot \vec{a}) \vec{r} - (\vec{d} \cdot \vec{r}) \vec{a} \]
Thus, we have:
\[ (\vec{d} \cdot \vec{a}) \vec{r} - (\vec{d} \cdot \vec{r}) \vec{a} = \vec{d} \times \vec{b} \]

### Step 3: Rearranging the equation
Rearranging gives us:
\[ (\vec{d} \cdot \vec{a}) \vec{r} - (\vec{d} \cdot \vec{r}) \vec{a} - \vec{d} \times \vec{b} = 0 \]

### Step 4: Cross the second equation with \( \vec{b} \)
Now, we take the second equation:
\[ \vec{r} \times \vec{c} = \vec{d} \]
Cross both sides with \( \vec{b} \):
\[ \vec{b} \times (\vec{r} \times \vec{c}) = \vec{b} \times \vec{d} \]

### Step 5: Apply the vector triple product identity again
Using the vector triple product identity again:
\[ \vec{b} \times (\vec{r} \times \vec{c}) = (\vec{b} \cdot \vec{c}) \vec{r} - (\vec{b} \cdot \vec{r}) \vec{c} \]
Thus, we have:
\[ (\vec{b} \cdot \vec{c}) \vec{r} - (\vec{b} \cdot \vec{r}) \vec{c} = \vec{b} \times \vec{d} \]

### Step 6: Rearranging the second equation
Rearranging gives us:
\[ (\vec{b} \cdot \vec{c}) \vec{r} - (\vec{b} \cdot \vec{r}) \vec{c} - \vec{b} \times \vec{d} = 0 \]

### Step 7: Combine both equations
Now we have two equations:
1. \((\vec{d} \cdot \vec{a}) \vec{r} - (\vec{d} \cdot \vec{r}) \vec{a} - \vec{d} \times \vec{b} = 0\)
2. \((\vec{b} \cdot \vec{c}) \vec{r} - (\vec{b} \cdot \vec{r}) \vec{c} - \vec{b} \times \vec{d} = 0\)

### Step 8: Set the coefficients of \( \vec{r} \) to be equal
For these equations to be consistent, the coefficients of \( \vec{r} \) must be equal. Therefore, we can conclude:
\[ \vec{b} \cdot \vec{c} + \vec{d} \cdot \vec{a} = 0 \]

### Final Condition
Thus, the condition for the equations to be consistent is:
\[ \vec{b} \cdot \vec{c} + \vec{d} \cdot \vec{a} = 0 \]