The vector `r = a xx [ a xx { a xx a xx (a xx b)}]` where `|a| = 2, |b| =5`,and a `bot` b forms an orthonormal system of vectors with a and b.
True or False?

Answer

To determine the truth of the statement regarding the vector \( r = a \times [ a \times ( a \times ( a \times b))] \), we will analyze it step by step.

### Step 1: Understand the Cross Product
The cross product of two vectors \( a \) and \( b \) is defined as:
\[ |a \times b| = |a||b|\sin(\theta) \]
where \( \theta \) is the angle between the two vectors. Since \( |a| = 2 \) and \( |b| = 5 \), and \( a \) and \( b \) are orthogonal (i.e., \( \theta = 90^\circ \)), we have:
\[ |a \times b| = |a||b|\sin(90^\circ) = 2 \times 5 \times 1 = 10 \]

### Step 2: Calculate \( a \times a \)
The cross product of any vector with itself is zero:
\[ a \times a = 0 \]

### Step 3: Substitute into the Expression
Now, substituting \( a \times a \) into the expression for \( r \):
\[ r = a \times [ a \times ( a \times ( a \times b))] \]
Since \( a \times a = 0 \), we can replace the first \( a \times a \) in the expression:
\[ r = a \times [ a \times (0)] \]

### Step 4: Calculate \( a \times 0 \)
The cross product of any vector with the zero vector is also zero:
\[ a \times 0 = 0 \]

### Step 5: Substitute Again
Now, substituting back into the expression for \( r \):
\[ r = a \times 0 = 0 \]

### Step 6: Conclusion
Since \( r = 0 \), it cannot form an orthonormal system with any vectors, including \( a \) and \( b \). Therefore, the statement is **False**.

### Final Answer
The statement is **False**.