Obtain the dot product of the vectors :
`vec(a)=hat(i)-hat(j)+hat(k)` and `vec(b)=hat(i)-hat(k)`.

Answer

To find the dot product of the vectors \(\vec{a} = \hat{i} - \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} - \hat{k}\), we can follow these steps:

### Step 1: Identify the components of the vectors
The vector \(\vec{a}\) can be expressed in terms of its components:
\[ \vec{a} = 1\hat{i} - 1\hat{j} + 1\hat{k} \]
So, the components are:
- \(x_1 = 1\) (coefficient of \(\hat{i}\))
- \(y_1 = -1\) (coefficient of \(\hat{j}\))
- \(z_1 = 1\) (coefficient of \(\hat{k}\))

For vector \(\vec{b}\):
\[ \vec{b} = 1\hat{i} + 0\hat{j} - 1\hat{k} \]
The components are:
- \(x_2 = 1\) (coefficient of \(\hat{i}\))
- \(y_2 = 0\) (coefficient of \(\hat{j}\))
- \(z_2 = -1\) (coefficient of \(\hat{k}\))

### Step 2: Apply the formula for the dot product
The dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is given by:
\[ \vec{a} \cdot \vec{b} = x_1 x_2 + y_1 y_2 + z_1 z_2 \]

### Step 3: Substitute the components into the formula
Now substituting the values:
\[ \vec{a} \cdot \vec{b} = (1)(1) + (-1)(0) + (1)(-1) \]

### Step 4: Calculate each term
Calculating each term:
1. \(1 \cdot 1 = 1\)
2. \(-1 \cdot 0 = 0\)
3. \(1 \cdot -1 = -1\)

Now, adding these results together:
\[ \vec{a} \cdot \vec{b} = 1 + 0 - 1 \]

### Step 5: Simplify the result
\[ \vec{a} \cdot \vec{b} = 1 - 1 = 0 \]

### Final Answer
The dot product of the vectors \(\vec{a}\) and \(\vec{b}\) is:
\[ \vec{a} \cdot \vec{b} = 0 \]