If `hat(i), hat(j), hat(k)` are the unit vectors and mutually perpendicular, then `[[hat(i), hat(j), hat(k)]]`=

Options

A. `0`
B. `-1`
C. `1`
D. `2`

Answer

To solve the problem, we need to find the scalar triple product of the unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\). The scalar triple product can be calculated using the formula:

\[ \text{Scalar Triple Product} = \hat{a} \cdot (\hat{b} \times \hat{c}) \]

where \(\hat{a}\), \(\hat{b}\), and \(\hat{c}\) are vectors. In our case, we can set:

\[ \hat{a} = \hat{i}, \quad \hat{b} = \hat{j}, \quad \hat{c} = \hat{k} \]

Thus, we can express the scalar triple product as:

\[ \hat{i} \cdot (\hat{j} \times \hat{k}) \]

### Step 1: Calculate the Cross Product \(\hat{j} \times \hat{k}\)

The cross product of two unit vectors that are mutually perpendicular is equal to the third unit vector. Therefore:

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

### Step 2: Substitute the Cross Product into the Scalar Triple Product

Now, we can substitute the result of the cross product back into our scalar triple product:

\[ \hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} \]

### Step 3: Calculate the Dot Product \(\hat{i} \cdot \hat{i}\)

The dot product of a unit vector with itself is equal to 1:

\[ \hat{i} \cdot \hat{i} = 1 \]

### Conclusion

Thus, the scalar triple product of \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) is:

\[ \hat{i} \cdot (\hat{j} \times \hat{k}) = 1 \]

The answer corresponds to option (c) which is 1.

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