Prove that `[hati" "hatj" "hatk]=1`, Hence give geometric interpretation.

Answer

To prove that the scalar triple product \([ \hat{i}, \hat{j}, \hat{k} ] = 1\), we will follow these steps:

### Step 1: Understand the Scalar Triple Product
The scalar triple product of three vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) is defined as:
\[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]
In our case, we will use the unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\).

### Step 2: Compute the Cross Product \(\hat{j} \times \hat{k}\)
We start by calculating the cross product of \(\hat{j}\) and \(\hat{k}\):
\[ \hat{j} \times \hat{k} = \hat{i} \]
This is a standard result from vector cross product properties.

### Step 3: Compute the Dot Product \(\hat{i} \cdot (\hat{j} \times \hat{k})\)
Next, we take the dot product of \(\hat{i}\) with the result from the previous step:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} = 1 \]
Since the dot product of a unit vector with itself is 1.

### Step 4: Conclusion
Thus, we have shown that:
\[ [\hat{i}, \hat{j}, \hat{k}] = \hat{i} \cdot (\hat{j} \times \hat{k}) = 1 \]

### Geometric Interpretation
The scalar triple product gives us the volume of the parallelepiped formed by the three vectors. In this case, since \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) are the unit vectors along the x, y, and z axes respectively, the volume of the unit cube formed by these vectors is 1.