If `veca=2hati+hatj+2hatk` then the value of `|hati xx(veca xxhati)|^2+|hatj xx(veca xx hatj)|^2+|hatk xx(veca xx hatk)|^2` is

Answer

To solve the given problem, we need to compute the expression:

\[ | \hat{i} \times (\vec{a} \times \hat{i}) |^2 + | \hat{j} \times (\vec{a} \times \hat{j}) |^2 + | \hat{k} \times (\vec{a} \times \hat{k}) |^2 \]

where \(\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}\).

### Step 1: Calculate \(\vec{a} \times \hat{i}\)

Using the formula for the cross product, we have:

\[ \vec{a} \times \hat{i} = (2\hat{i} + \hat{j} + 2\hat{k}) \times \hat{i} \]

Calculating this, we get:

\[ \vec{a} \times \hat{i} = \hat{j} \times \hat{i} + 2\hat{k} \times \hat{i} = -\hat{k} + 2(-\hat{j}) = -\hat{k} - 2\hat{j} \]

### Step 2: Calculate \(\hat{i} \times (\vec{a} \times \hat{i})\)

Now we need to compute:

\[ \hat{i} \times (\vec{a} \times \hat{i}) = \hat{i} \times (-\hat{k} - 2\hat{j}) \]

Using the distributive property of the cross product:

\[ \hat{i} \times (-\hat{k}) + \hat{i} \times (-2\hat{j}) = -\hat{i} \times \hat{k} - 2\hat{i} \times \hat{j} = -(-\hat{j}) - 2\hat{k} = \hat{j} - 2\hat{k} \]

### Step 3: Calculate the magnitude squared

Now we calculate the magnitude squared:

\[ | \hat{i} \times (\vec{a} \times \hat{i}) |^2 = | \hat{j} - 2\hat{k} |^2 = 1^2 + (-2)^2 = 1 + 4 = 5 \]

### Step 4: Calculate \(\vec{a} \times \hat{j}\)

Next, we compute:

\[ \vec{a} \times \hat{j} = (2\hat{i} + \hat{j} + 2\hat{k}) \times \hat{j} \]

Calculating this, we get:

\[ \vec{a} \times \hat{j} = 2\hat{i} \times \hat{j} + 2\hat{k} \times \hat{j} = 2\hat{k} - 2\hat{i} \]

### Step 5: Calculate \(\hat{j} \times (\vec{a} \times \hat{j})\)

Now we compute:

\[ \hat{j} \times (\vec{a} \times \hat{j}) = \hat{j} \times (2\hat{k} - 2\hat{i}) = 2(\hat{j} \times \hat{k}) - 2(\hat{j} \times \hat{i}) = 2(-\hat{i}) - 2\hat{k} = -2\hat{i} - 2\hat{k} \]

### Step 6: Calculate the magnitude squared

Now we calculate the magnitude squared:

\[ | \hat{j} \times (\vec{a} \times \hat{j}) |^2 = | -2\hat{i} - 2\hat{k} |^2 = (-2)^2 + 0^2 + (-2)^2 = 4 + 0 + 4 = 8 \]

### Step 7: Calculate \(\vec{a} \times \hat{k}\)

Next, we compute:

\[ \vec{a} \times \hat{k} = (2\hat{i} + \hat{j} + 2\hat{k}) \times \hat{k} \]

Calculating this, we get:

\[ \vec{a} \times \hat{k} = 2\hat{i} \times \hat{k} + \hat{j} \times \hat{k} = -2\hat{j} + \hat{i} \]

### Step 8: Calculate \(\hat{k} \times (\vec{a} \times \hat{k})\)

Now we compute:

\[ \hat{k} \times (\vec{a} \times \hat{k}) = \hat{k} \times (-2\hat{j} + \hat{i}) = -2(\hat{k} \times \hat{j}) + \hat{k} \times \hat{i} = -2(-\hat{i}) + \hat{j} = 2\hat{i} + \hat{j} \]

### Step 9: Calculate the magnitude squared

Now we calculate the magnitude squared:

\[ | \hat{k} \times (\vec{a} \times \hat{k}) |^2 = | 2\hat{i} + \hat{j} |^2 = 2^2 + 1^2 = 4 + 1 = 5 \]

### Step 10: Combine all the results

Finally, we combine all the results:

\[ | \hat{i} \times (\vec{a} \times \hat{i}) |^2 + | \hat{j} \times (\vec{a} \times \hat{j}) |^2 + | \hat{k} \times (\vec{a} \times \hat{k}) |^2 = 5 + 8 + 5 = 18 \]

Thus, the value of the expression is:

\[ \boxed{18} \]