Find `vec(a).vec(b)` when
(i) `vec(a)=hat(i)-2hat(j)+hat(k)` and `vec(b)=3 hat(i)-4 hat(j)-2 hat(k)`
(ii) `vec(a)=hat(i)+2hat(j)+3hat(k)` and `vec(b)=-2hat(j)+4hat(k)`
(iii) `vec(a)=hat(i)-hat(j)+5hat(k)` and `vec(b)=3 hat(i)-2 hat(k)`

Answer

To find the dot product of the vectors \(\vec{a}\) and \(\vec{b}\), we will use the formula for the dot product:

\[ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \]

where \(\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\) and \(\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}\).

### Part (i)
Given:
\[ \vec{a} = \hat{i} - 2\hat{j} + \hat{k} \quad \text{and} \quad \vec{b} = 3\hat{i} - 4\hat{j} - 2\hat{k} \]

1. Identify the components:
- \(a_1 = 1\), \(a_2 = -2\), \(a_3 = 1\)
- \(b_1 = 3\), \(b_2 = -4\), \(b_3 = -2\)

2. Apply the dot product formula:
\[ \vec{a} \cdot \vec{b} = (1)(3) + (-2)(-4) + (1)(-2) \]

3. Calculate each term:
- \(1 \cdot 3 = 3\)
- \(-2 \cdot -4 = 8\)
- \(1 \cdot -2 = -2\)

4. Sum the results:
\[ \vec{a} \cdot \vec{b} = 3 + 8 - 2 = 9 \]

### Part (ii)
Given:
\[ \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \quad \text{and} \quad \vec{b} = -2\hat{j} + 4\hat{k} \]

1. Identify the components:
- \(a_1 = 1\), \(a_2 = 2\), \(a_3 = 3\)
- \(b_1 = 0\), \(b_2 = -2\), \(b_3 = 4\) (note that \(b_1 = 0\) since there is no \(\hat{i}\) component)

2. Apply the dot product formula:
\[ \vec{a} \cdot \vec{b} = (1)(0) + (2)(-2) + (3)(4) \]

3. Calculate each term:
- \(1 \cdot 0 = 0\)
- \(2 \cdot -2 = -4\)
- \(3 \cdot 4 = 12\)

4. Sum the results:
\[ \vec{a} \cdot \vec{b} = 0 - 4 + 12 = 8 \]

### Part (iii)
Given:
\[ \vec{a} = \hat{i} - \hat{j} + 5\hat{k} \quad \text{and} \quad \vec{b} = 3\hat{i} - 2\hat{k} \]

1. Identify the components:
- \(a_1 = 1\), \(a_2 = -1\), \(a_3 = 5\)
- \(b_1 = 3\), \(b_2 = 0\), \(b_3 = -2\) (note that \(b_2 = 0\) since there is no \(\hat{j}\) component)

2. Apply the dot product formula:
\[ \vec{a} \cdot \vec{b} = (1)(3) + (-1)(0) + (5)(-2) \]

3. Calculate each term:
- \(1 \cdot 3 = 3\)
- \(-1 \cdot 0 = 0\)
- \(5 \cdot -2 = -10\)

4. Sum the results:
\[ \vec{a} \cdot \vec{b} = 3 + 0 - 10 = -7 \]

### Summary of Results:
- (i) \(\vec{a} \cdot \vec{b} = 9\)
- (ii) \(\vec{a} \cdot \vec{b} = 8\)
- (iii) \(\vec{a} \cdot \vec{b} = -7\)