If r be any vector, then
`|r xx i|^(2) + |r xx j|^(2) + |r xxk|^(2) =` ........

Answer

To solve the problem, we need to find the expression \( |r \times \hat{i}|^2 + |r \times \hat{j}|^2 + |r \times \hat{k}|^2 \) where \( r \) is a vector represented as \( r = r_1 \hat{i} + r_2 \hat{j} + r_3 \hat{k} \).

### Step-by-Step Solution:

1. **Define the Vector**:
Let \( r = r_1 \hat{i} + r_2 \hat{j} + r_3 \hat{k} \).

2. **Calculate \( r \times \hat{i} \)**:
\[ r \times \hat{i} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_1 & r_2 & r_3 \\ 1 & 0 & 0 \end{vmatrix} \]
Expanding this determinant:
\[ = \hat{i}(r_2 \cdot 0 - r_3 \cdot 0) - \hat{j}(r_1 \cdot 0 - r_3 \cdot 1) + \hat{k}(r_1 \cdot 0 - r_2 \cdot 1) \]
\[ = -r_3 \hat{j} + r_2 \hat{k} \]
Therefore,
\[ r \times \hat{i} = -r_3 \hat{j} + r_2 \hat{k} \]

3. **Magnitude of \( r \times \hat{i} \)**:
\[ |r \times \hat{i}|^2 = (-r_3)^2 + (r_2)^2 = r_3^2 + r_2^2 \]

4. **Calculate \( r \times \hat{j} \)**:
\[ r \times \hat{j} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_1 & r_2 & r_3 \\ 0 & 1 & 0 \end{vmatrix} \]
Expanding this determinant:
\[ = \hat{i}(r_2 \cdot 0 - r_3 \cdot 0) - \hat{j}(r_1 \cdot 0 - r_3 \cdot 1) + \hat{k}(r_1 \cdot 1 - r_2 \cdot 0) \]
\[ = r_3 \hat{i} - r_1 \hat{k} \]
Therefore,
\[ r \times \hat{j} = r_3 \hat{i} - r_1 \hat{k} \]

5. **Magnitude of \( r \times \hat{j} \)**:
\[ |r \times \hat{j}|^2 = (r_3)^2 + (-r_1)^2 = r_3^2 + r_1^2 \]

6. **Calculate \( r \times \hat{k} \)**:
\[ r \times \hat{k} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_1 & r_2 & r_3 \\ 0 & 0 & 1 \end{vmatrix} \]
Expanding this determinant:
\[ = \hat{i}(r_2 \cdot 1 - r_3 \cdot 0) - \hat{j}(r_1 \cdot 1 - r_3 \cdot 0) + \hat{k}(r_1 \cdot 0 - r_2 \cdot 0) \]
\[ = r_2 \hat{i} - r_1 \hat{j} \]
Therefore,
\[ r \times \hat{k} = r_2 \hat{i} - r_1 \hat{j} \]

7. **Magnitude of \( r \times \hat{k} \)**:
\[ |r \times \hat{k}|^2 = (r_2)^2 + (-r_1)^2 = r_2^2 + r_1^2 \]

8. **Final Calculation**:
Now, we sum the magnitudes:
\[ |r \times \hat{i}|^2 + |r \times \hat{j}|^2 + |r \times \hat{k}|^2 = (r_3^2 + r_2^2) + (r_3^2 + r_1^2) + (r_2^2 + r_1^2) \]
\[ = 2r_1^2 + 2r_2^2 + 2r_3^2 \]
\[ = 2(r_1^2 + r_2^2 + r_3^2) \]

9. **Magnitude of Vector \( r \)**:
The magnitude of vector \( r \) is given by:
\[ |r|^2 = r_1^2 + r_2^2 + r_3^2 \]

10. **Final Result**:
Therefore, the final result is:
\[ |r \times \hat{i}|^2 + |r \times \hat{j}|^2 + |r \times \hat{k}|^2 = 2|r|^2 \]