The value of
`abs(a times(i timesj))^(2)+abs(a times(j timesk))^(2)+abs(a times(k times i))^(2)` is

Options

A. `abs(a)^(2)`
B. `2abs(a)^(2)`
C. `3abs(a)^(2)`
D. none of these

Answer

To solve the problem, we need to evaluate the expression:

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

Let’s denote the vector \(\mathbf{a}\) as:

\[ \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \]

### Step 1: Calculate \(\mathbf{i} \times \mathbf{j}\)

Using the right-hand rule for cross products:

\[ \mathbf{i} \times \mathbf{j} = \mathbf{k} \]

### Step 2: Calculate \(\mathbf{a} \times (\mathbf{i} \times \mathbf{j})\)

Substituting the result from Step 1:

\[ \mathbf{a} \times \mathbf{k} = (a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}) \times \mathbf{k} \]

Using the properties of cross products:

\[ = a_x (\mathbf{i} \times \mathbf{k}) + a_y (\mathbf{j} \times \mathbf{k}) + a_z (\mathbf{k} \times \mathbf{k}) \]

Calculating each term:

- \(\mathbf{i} \times \mathbf{k} = -\mathbf{j}\)
- \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\)
- \(\mathbf{k} \times \mathbf{k} = \mathbf{0}\)

Thus, we have:

\[ \mathbf{a} \times \mathbf{k} = a_x (-\mathbf{j}) + a_y \mathbf{i} + 0 = a_y \mathbf{i} - a_x \mathbf{j} \]

### Step 3: Calculate the magnitude squared

Now, we find the magnitude squared:

\[ | \mathbf{a} \times (\mathbf{i} \times \mathbf{j}) |^2 = | a_y \mathbf{i} - a_x \mathbf{j} |^2 = a_y^2 + a_x^2 \]

### Step 4: Calculate \(\mathbf{j} \times \mathbf{k}\)

Using the right-hand rule:

\[ \mathbf{j} \times \mathbf{k} = \mathbf{i} \]

### Step 5: Calculate \(\mathbf{a} \times (\mathbf{j} \times \mathbf{k})\)

Substituting the result from Step 4:

\[ \mathbf{a} \times \mathbf{i} = (a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}) \times \mathbf{i} \]

Calculating each term:

- \(\mathbf{j} \times \mathbf{i} = -\mathbf{k}\)
- \(\mathbf{k} \times \mathbf{i} = \mathbf{j}\)

Thus, we have:

\[ \mathbf{a} \times \mathbf{i} = a_y (-\mathbf{k}) + a_z \mathbf{j} + 0 = a_z \mathbf{j} - a_y \mathbf{k} \]

### Step 6: Calculate the magnitude squared

Now, we find the magnitude squared:

\[ | \mathbf{a} \times (\mathbf{j} \times \mathbf{k}) |^2 = | a_z \mathbf{j} - a_y \mathbf{k} |^2 = a_z^2 + a_y^2 \]

### Step 7: Calculate \(\mathbf{k} \times \mathbf{i}\)

Using the right-hand rule:

\[ \mathbf{k} \times \mathbf{i} = -\mathbf{j} \]

### Step 8: Calculate \(\mathbf{a} \times (\mathbf{k} \times \mathbf{i})\)

Substituting the result from Step 7:

\[ \mathbf{a} \times (-\mathbf{j}) = - (a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}) \times \mathbf{j} \]

Calculating each term:

- \(\mathbf{i} \times \mathbf{j} = \mathbf{k}\)
- \(\mathbf{j} \times \mathbf{j} = \mathbf{0}\)
- \(\mathbf{k} \times \mathbf{j} = -\mathbf{i}\)

Thus, we have:

\[ \mathbf{a} \times (-\mathbf{j}) = - a_x \mathbf{k} + 0 + a_z \mathbf{i} = a_z \mathbf{i} - a_x \mathbf{k} \]

### Step 9: Calculate the magnitude squared

Now, we find the magnitude squared:

\[ | \mathbf{a} \times (\mathbf{k} \times \mathbf{i}) |^2 = | a_z \mathbf{i} - a_x \mathbf{k} |^2 = a_z^2 + a_x^2 \]

### Step 10: Combine all results

Now we can combine all the results:

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

Substituting the values we calculated:

\[ = (a_y^2 + a_x^2) + (a_z^2 + a_y^2) + (a_z^2 + a_x^2) \]

Combining like terms:

\[ = 2(a_x^2 + a_y^2 + a_z^2) \]

### Final Result

Thus, the final value is:

\[ = 2 |\mathbf{a}|^2 \]