`a*((b times c) times (a+(b times c))` is equal to

Options

A. 0
B. 2 [a b c]
C. [a b c]
D. none of these

Answer

To solve the expression \( a \cdot ((b \times c) \times (a + (b \times c))) \), we will follow these steps:

### Step 1: Apply the Distributive Law
We can rewrite the expression using the distributive property of the cross product:
\[ a \cdot ((b \times c) \times (a + (b \times c))) = a \cdot ((b \times c) \times a) + a \cdot ((b \times c) \times (b \times c)) \]

### Step 2: Simplify the Second Term
The second term \( (b \times c) \times (b \times c) \) is equal to zero because the cross product of any vector with itself is zero:
\[ (b \times c) \times (b \times c) = 0 \]
Thus, we can simplify our expression to:
\[ a \cdot ((b \times c) \times a) + 0 = a \cdot ((b \times c) \times a) \]

### Step 3: Use the Vector Triple Product Identity
We can apply the vector triple product identity, which states:
\[ x \times (y \times z) = (x \cdot z)y - (x \cdot y)z \]
In our case, let \( x = a \), \( y = b \), and \( z = c \):
\[ (b \times c) \times a = (b \cdot a)c - (c \cdot a)b \]
So, substituting this back into our expression gives:
\[ a \cdot ((b \cdot a)c - (c \cdot a)b) \]

### Step 4: Distribute the Dot Product
Now, we distribute the dot product:
\[ a \cdot ((b \cdot a)c) - a \cdot ((c \cdot a)b) \]
This can be simplified as:
\[ (b \cdot a)(a \cdot c) - (c \cdot a)(a \cdot b) \]

### Step 5: Recognize the Terms
Notice that \( b \cdot a \) and \( a \cdot b \) are equal, and \( c \cdot a \) and \( a \cdot c \) are also equal:
\[ (b \cdot a)(a \cdot c) - (c \cdot a)(b \cdot a) \]
This means:
\[ x \cdot y - x \cdot y = 0 \]
Thus, the entire expression simplifies to:
\[ 0 \]

### Final Answer
Therefore, the value of the expression \( a \cdot ((b \times c) \times (a + (b \times c))) \) is:
\[ \boxed{0} \]

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