`(bara times hati)^2+(bara+hatj)^2+(bara times hatk)^2` is equal to

Options

A. `(bara)^2`
B. `3(bara)^2`
C. `2(bara)^2`
D. None of these

Answer

To solve the expression \((\mathbf{a} \times \hat{i})^2 + (\mathbf{a} \times \hat{j})^2 + (\mathbf{a} \times \hat{k})^2\), we will follow these steps:

### Step 1: Define the vector \(\mathbf{a}\)
Let \(\mathbf{a} = x \hat{i} + y \hat{j} + z \hat{k}\), where \(x\), \(y\), and \(z\) are the components of the vector \(\mathbf{a}\).

### Step 2: Calculate \(\mathbf{a} \times \hat{i}\)
Using the determinant method for the cross product:
\[ \mathbf{a} \times \hat{i} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x & y & z \\ 1 & 0 & 0 \end{vmatrix} \]
Calculating the determinant, we get:
\[ \mathbf{a} \times \hat{i} = \hat{i}(y \cdot 0 - z \cdot 0) - \hat{j}(x \cdot 0 - z \cdot 1) + \hat{k}(x \cdot 0 - y \cdot 1) = -z \hat{j} + y \hat{k} \]
Thus,
\[ \mathbf{a} \times \hat{i} = z \hat{j} + y \hat{k} \]

### Step 3: Calculate \((\mathbf{a} \times \hat{i})^2\)
Now, we find the magnitude squared:
\[ (\mathbf{a} \times \hat{i})^2 = |z \hat{j} + y \hat{k}|^2 = z^2 + y^2 \]

### Step 4: Calculate \(\mathbf{a} \times \hat{j}\)
Similarly, we calculate:
\[ \mathbf{a} \times \hat{j} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x & y & z \\ 0 & 1 & 0 \end{vmatrix} \]
Calculating the determinant, we get:
\[ \mathbf{a} \times \hat{j} = z \hat{i} - x \hat{k} \]
Thus,
\[ \mathbf{a} \times \hat{j} = z \hat{i} - x \hat{k} \]

### Step 5: Calculate \((\mathbf{a} \times \hat{j})^2\)
Magnitude squared:
\[ (\mathbf{a} \times \hat{j})^2 = |z \hat{i} - x \hat{k}|^2 = z^2 + x^2 \]

### Step 6: Calculate \(\mathbf{a} \times \hat{k}\)
Now, we calculate:
\[ \mathbf{a} \times \hat{k} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x & y & z \\ 0 & 0 & 1 \end{vmatrix} \]
Calculating the determinant, we get:
\[ \mathbf{a} \times \hat{k} = -y \hat{i} + x \hat{j} \]
Thus,
\[ \mathbf{a} \times \hat{k} = -y \hat{i} + x \hat{j} \]

### Step 7: Calculate \((\mathbf{a} \times \hat{k})^2\)
Magnitude squared:
\[ (\mathbf{a} \times \hat{k})^2 = |-y \hat{i} + x \hat{j}|^2 = y^2 + x^2 \]

### Step 8: Combine the results
Now we combine all three results:
\[ (\mathbf{a} \times \hat{i})^2 + (\mathbf{a} \times \hat{j})^2 + (\mathbf{a} \times \hat{k})^2 = (z^2 + y^2) + (z^2 + x^2) + (y^2 + x^2) \]
This simplifies to:
\[ = 2x^2 + 2y^2 + 2z^2 = 2(x^2 + y^2 + z^2) \]

### Final Result
Thus, the final answer is:
\[ = 2|\mathbf{a}|^2 \]