Given two orthogonal vectors `vec(A)` and `vec(B)` each of length unity . Let `vec(P)` be the vector satisfying the equation `vec(P) xx vec(B) = vec(A) - vec(P)` . Then
`(vec(P) xx vec(B)) xx vec(B)` is equal to .

Answer

To solve the problem step by step, we start with the given information and the equation involving the vectors.

### Step 1: Understand the Given Information
We have two orthogonal unit vectors \(\vec{A}\) and \(\vec{B}\). This means:
- \(|\vec{A}| = 1\)
- \(|\vec{B}| = 1\)
- \(\vec{A} \cdot \vec{B} = 0\) (since they are orthogonal)

### Step 2: Write Down the Given Equation
We are given the equation:
\[ \vec{P} \times \vec{B} = \vec{A} - \vec{P} \]

### Step 3: Cross Product with \(\vec{B}\)
We will take the cross product of both sides of the equation with \(\vec{B}\):
\[ (\vec{P} \times \vec{B}) \times \vec{B} = (\vec{A} - \vec{P}) \times \vec{B} \]

### Step 4: Use the Vector Triple Product Identity
Using the vector triple product identity, we can simplify the left side:
\[ (\vec{P} \times \vec{B}) \times \vec{B} = \vec{P}(\vec{B} \cdot \vec{B}) - \vec{B}(\vec{P} \cdot \vec{B}) \]
Since \(|\vec{B}| = 1\), we have \(\vec{B} \cdot \vec{B} = 1\):
\[ (\vec{P} \times \vec{B}) \times \vec{B} = \vec{P} - \vec{B}(\vec{P} \cdot \vec{B}) \]

### Step 5: Simplify the Right Side
Now, we need to simplify the right side:
\[ (\vec{A} - \vec{P}) \times \vec{B} = \vec{A} \times \vec{B} - \vec{P} \times \vec{B} \]
Substituting \(\vec{P} \times \vec{B} = \vec{A} - \vec{P}\) into this gives:
\[ \vec{A} \times \vec{B} - (\vec{A} - \vec{P}) = \vec{A} \times \vec{B} - \vec{A} + \vec{P} \]

### Step 6: Set the Two Sides Equal
Now we have:
\[ \vec{P} - \vec{B}(\vec{P} \cdot \vec{B}) = \vec{A} \times \vec{B} - \vec{A} + \vec{P} \]
Subtract \(\vec{P}\) from both sides:
\[ -\vec{B}(\vec{P} \cdot \vec{B}) = \vec{A} \times \vec{B} - \vec{A} \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ \vec{B}(\vec{P} \cdot \vec{B}) = \vec{A} - \vec{A} \times \vec{B} \]

### Step 8: Find \(\vec{P} \cdot \vec{B}\)
Since \(\vec{A}\) and \(\vec{B}\) are orthogonal, \(\vec{A} \times \vec{B}\) is perpendicular to both \(\vec{A}\) and \(\vec{B}\). Thus, we can conclude that:
\[ \vec{P} \cdot \vec{B} = 0 \]

### Step 9: Final Result
Substituting \(\vec{P} \cdot \vec{B} = 0\) back into the equation gives:
\[ \vec{P} \times \vec{B} \times \vec{B} = \vec{A} \times \vec{B} - \vec{A} \]

### Conclusion
Thus, the final expression for \((\vec{P} \times \vec{B}) \times \vec{B}\) is:
\[ \vec{A} \times \vec{B} - \vec{A} \]