For any vector `vec(a)`, the value of ` |vec(a) xx hat(i)|^(2) + |vec(a) xx hat(j)|^(2) + |vec(a) xx hat(k)|^(2)` is equal to

Options

A. `|vec(a)|^(2)`
B. `2|vec(a)|^(2)`
C. `3|vec(a)|^(2)`
D. `4|vec(a)|^(2)`

Answer

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

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

Let's denote the vector \(\vec{a}\) as:

\[ \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \]

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

Using the properties of the cross product, we can calculate:

\[ \vec{a} \times \hat{i} = (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) \times \hat{i} \]

Using the right-hand rule and the properties of the cross product:

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

Thus, we have:

\[ \vec{a} \times \hat{i} = a_2 \hat{j} \times \hat{i} + a_3 \hat{k} \times \hat{i} = -a_2 \hat{k} + a_3 \hat{j} \]

So:

\[ \vec{a} \times \hat{i} = -a_2 \hat{k} + a_3 \hat{j} \]

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

Now, we find the magnitude squared:

\[ |\vec{a} \times \hat{i}|^2 = |-a_2 \hat{k} + a_3 \hat{j}|^2 = a_2^2 + a_3^2 \]

### Step 3: Calculate \(\vec{a} \times \hat{j}\)

Next, we compute:

\[ \vec{a} \times \hat{j} = (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) \times \hat{j} \]

Using the properties of the cross product:

\[ \vec{a} \times \hat{j} = a_1 \hat{i} \times \hat{j} + a_3 \hat{k} \times \hat{j} = a_1 \hat{k} - a_3 \hat{i} \]

### Step 4: Calculate \(|\vec{a} \times \hat{j}|^2\)

Now we find the magnitude squared:

\[ |\vec{a} \times \hat{j}|^2 = |a_1 \hat{k} - a_3 \hat{i}|^2 = a_1^2 + a_3^2 \]

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

Now we compute:

\[ \vec{a} \times \hat{k} = (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) \times \hat{k} \]

Using the properties of the cross product:

\[ \vec{a} \times \hat{k} = a_1 \hat{i} \times \hat{k} + a_2 \hat{j} \times \hat{k} = -a_1 \hat{j} + a_2 \hat{i} \]

### Step 6: Calculate \(|\vec{a} \times \hat{k}|^2\)

Now we find the magnitude squared:

\[ |\vec{a} \times \hat{k}|^2 = |-a_1 \hat{j} + a_2 \hat{i}|^2 = a_1^2 + a_2^2 \]

### Step 7: Combine the results

Now we can combine all the results:

\[ |\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2 = (a_2^2 + a_3^2) + (a_1^2 + a_3^2) + (a_1^2 + a_2^2) \]

This simplifies to:

\[ = 2(a_1^2 + a_2^2 + a_3^2) \]

### Step 8: Relate to \(|\vec{a}|^2\)

Since \(|\vec{a}|^2 = a_1^2 + a_2^2 + a_3^2\), we can express the result as:

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

### Final Result

Thus, the final answer is:

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

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