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

Options

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

Answer

To solve the expression \( \text{abs}(a \times \mathbf{i})^2 + \text{abs}(a \times \mathbf{j})^2 + \text{abs}(a \times \mathbf{k})^2 \), we will break it down step by step.

### Step 1: Define the vector \( \mathbf{a} \)
Let the vector \( \mathbf{a} \) be represented in component form:
\[ \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \]

### Step 2: Calculate \( \mathbf{a} \times \mathbf{i} \)
Using the properties of the cross product:
\[ \mathbf{a} \times \mathbf{i} = (a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}) \times \mathbf{i} \]
Applying the cross product:
- \( \mathbf{i} \times \mathbf{i} = \mathbf{0} \)
- \( \mathbf{j} \times \mathbf{i} = -\mathbf{k} \)
- \( \mathbf{k} \times \mathbf{i} = \mathbf{j} \)

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

### Step 3: Calculate the magnitude squared of \( \mathbf{a} \times \mathbf{i} \)
Now, we find the magnitude squared:
\[ |\mathbf{a} \times \mathbf{i}|^2 = |a_z \mathbf{j} - a_y \mathbf{k}|^2 = (0)^2 + (a_z)^2 + (-a_y)^2 = a_y^2 + a_z^2 \]

### Step 4: Calculate \( \mathbf{a} \times \mathbf{j} \)
Now, we calculate \( \mathbf{a} \times \mathbf{j} \):
\[ \mathbf{a} \times \mathbf{j} = (a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}) \times \mathbf{j} \]
Applying the cross product:
- \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \)
- \( \mathbf{j} \times \mathbf{j} = \mathbf{0} \)
- \( \mathbf{k} \times \mathbf{j} = -\mathbf{i} \)

Thus,
\[ \mathbf{a} \times \mathbf{j} = a_x \mathbf{k} - a_z \mathbf{i} \]

### Step 5: Calculate the magnitude squared of \( \mathbf{a} \times \mathbf{j} \)
Now, we find the magnitude squared:
\[ |\mathbf{a} \times \mathbf{j}|^2 = |a_x \mathbf{k} - a_z \mathbf{i}|^2 = (0)^2 + (-a_z)^2 + (a_x)^2 = a_x^2 + a_z^2 \]

### Step 6: Calculate \( \mathbf{a} \times \mathbf{k} \)
Now, we calculate \( \mathbf{a} \times \mathbf{k} \):
\[ \mathbf{a} \times \mathbf{k} = (a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}) \times \mathbf{k} \]
Applying the cross product:
- \( \mathbf{i} \times \mathbf{k} = -\mathbf{j} \)
- \( \mathbf{j} \times \mathbf{k} = \mathbf{i} \)
- \( \mathbf{k} \times \mathbf{k} = \mathbf{0} \)

Thus,
\[ \mathbf{a} \times \mathbf{k} = -a_x \mathbf{j} + a_y \mathbf{i} \]

### Step 7: Calculate the magnitude squared of \( \mathbf{a} \times \mathbf{k} \)
Now, we find the magnitude squared:
\[ |\mathbf{a} \times \mathbf{k}|^2 = |-a_x \mathbf{j} + a_y \mathbf{i}|^2 = (a_y)^2 + (-a_x)^2 = a_x^2 + a_y^2 \]

### Step 8: Combine the results
Now, we sum the magnitudes squared:
\[ |\mathbf{a} \times \mathbf{i}|^2 + |\mathbf{a} \times \mathbf{j}|^2 + |\mathbf{a} \times \mathbf{k}|^2 = (a_y^2 + a_z^2) + (a_x^2 + a_z^2) + (a_x^2 + a_y^2) \]
This simplifies to:
\[ = 2(a_x^2 + a_y^2 + a_z^2) \]

### Step 9: Relate to the magnitude of \( \mathbf{a} \)
The magnitude of \( \mathbf{a} \) is given by:
\[ |\mathbf{a}|^2 = a_x^2 + a_y^2 + a_z^2 \]
Thus,
\[ 2(a_x^2 + a_y^2 + a_z^2) = 2|\mathbf{a}|^2 \]

### Final Answer
The value of \( \text{abs}(a \times \mathbf{i})^2 + \text{abs}(a \times \mathbf{j})^2 + \text{abs}(a \times \mathbf{k})^2 \) is:
\[ \boxed{2 |\mathbf{a}|^2} \]