If `4veca+5vecb+9vecc=0 " then " (vecaxxvecb)xx[(vecbxxvecc)xx(veccxxveca)]` is equal to

Options

A. a vector perpendicular to the plane of `veca, vecb and vecc`
B. a scalar quantity
C. `vec0`
D. none of these

Explanation

To solve the problem, we need to evaluate the expression \( \vec{a} \times (\vec{b} \times (\vec{c} \times \vec{a})) \) given that \( 4\vec{a} + 5\vec{b} + 9\vec{c} = 0 \).

### Step 1: Rewrite the given equation
We start with the equation:
\[ 4\vec{a} + 5\vec{b} + 9\vec{c} = 0 \]
From this, we can express one vector in terms of the others. For instance, we can express \( \vec{c} \):
\[ \vec{c} = -\frac{4}{9}\vec{a} - \frac{5}{9}\vec{b} \]

**Hint:** You can isolate one vector in terms of the others to simplify the expression.

### Step 2: Substitute \( \vec{c} \) into the expression
Now, we substitute \( \vec{c} \) into the expression we want to evaluate:
\[ \vec{a} \times (\vec{b} \times (\vec{c} \times \vec{a})) \]

Using the expression for \( \vec{c} \):
\[ \vec{c} \times \vec{a} = \left(-\frac{4}{9}\vec{a} - \frac{5}{9}\vec{b}\right) \times \vec{a} \]
Since the cross product of any vector with itself is zero, we have:
\[ \vec{c} \times \vec{a} = -\frac{4}{9}(\vec{a} \times \vec{a}) - \frac{5}{9}(\vec{b} \times \vec{a}) = 0 - \frac{5}{9}(\vec{b} \times \vec{a}) = -\frac{5}{9}(\vec{b} \times \vec{a}) \]

**Hint:** Remember that the cross product is anti-commutative, meaning \( \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b}) \).

### Step 3: Substitute back into the expression
Now we can substitute this result back into our original expression:
\[ \vec{a} \times (\vec{b} \times (-\frac{5}{9}(\vec{b} \times \vec{a}))) \]
This simplifies to:
\[ -\frac{5}{9} \vec{a} \times (\vec{b} \times (\vec{b} \times \vec{a})) \]

**Hint:** Use the vector triple product identity: \( \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z})\vec{y} - (\vec{x} \cdot \vec{y})\vec{z} \).

### Step 4: Apply the vector triple product identity
Applying the vector triple product identity:
\[ \vec{b} \times (\vec{b} \times \vec{a}) = (\vec{b} \cdot \vec{a})\vec{b} - (\vec{b} \cdot \vec{b})\vec{a} \]
Thus, we have:
\[ -\frac{5}{9} \vec{a} \times \left((\vec{b} \cdot \vec{a})\vec{b} - (\vec{b} \cdot \vec{b})\vec{a}\right) \]

**Hint:** Note that the cross product of a vector with itself results in zero.

### Step 5: Evaluate the final expression
The term \( \vec{a} \times \vec{a} \) is zero, so we only need to consider:
\[ -\frac{5}{9} (\vec{b} \cdot \vec{b}) (\vec{a} \times \vec{a}) = 0 \]

Thus, the entire expression simplifies to:
\[ 0 \]

### Final Answer
The value of \( \vec{a} \times (\vec{b} \times (\vec{c} \times \vec{a})) \) is:
\[ \boxed{0} \]