If [I, j, k] be a set of orthogonal unit vectors, then fill up the blanks :
(i) `i.i +j.j +k.k =.........`
(ii) `i.j+j.k+k.i. = ..........`
(iii) `i.i =j.j =k.k = `.........
(iv) `i.j=j.k =k.i. =` ........

Answer

To solve the given problem step by step, we will analyze each part of the question regarding the orthogonal unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).

### Step 1: Calculate \( \mathbf{i} \cdot \mathbf{i} + \mathbf{j} \cdot \mathbf{j} + \mathbf{k} \cdot \mathbf{k} \)

1. The dot product of a unit vector with itself is 1. Therefore:
- \( \mathbf{i} \cdot \mathbf{i} = 1 \)
- \( \mathbf{j} \cdot \mathbf{j} = 1 \)
- \( \mathbf{k} \cdot \mathbf{k} = 1 \)

2. Adding these values together:
\[ \mathbf{i} \cdot \mathbf{i} + \mathbf{j} \cdot \mathbf{j} + \mathbf{k} \cdot \mathbf{k} = 1 + 1 + 1 = 3 \]

**Answer for (i):** \( 3 \)

### Step 2: Calculate \( \mathbf{i} \cdot \mathbf{j} + \mathbf{j} \cdot \mathbf{k} + \mathbf{k} \cdot \mathbf{i} \)

1. The dot product of two orthogonal unit vectors is 0. Therefore:
- \( \mathbf{i} \cdot \mathbf{j} = 0 \)
- \( \mathbf{j} \cdot \mathbf{k} = 0 \)
- \( \mathbf{k} \cdot \mathbf{i} = 0 \)

2. Adding these values together:
\[ \mathbf{i} \cdot \mathbf{j} + \mathbf{j} \cdot \mathbf{k} + \mathbf{k} \cdot \mathbf{i} = 0 + 0 + 0 = 0 \]

**Answer for (ii):** \( 0 \)

### Step 3: Calculate \( \mathbf{i} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{j} = \mathbf{k} \cdot \mathbf{k} \)

1. As established earlier, the dot product of each unit vector with itself is:
- \( \mathbf{i} \cdot \mathbf{i} = 1 \)
- \( \mathbf{j} \cdot \mathbf{j} = 1 \)
- \( \mathbf{k} \cdot \mathbf{k} = 1 \)

2. Therefore, we can conclude:
\[ \mathbf{i} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{j} = \mathbf{k} \cdot \mathbf{k} = 1 \]

**Answer for (iii):** \( 1 \)

### Step 4: Calculate \( \mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} \)

1. As established earlier, the dot product of each pair of orthogonal unit vectors is:
- \( \mathbf{i} \cdot \mathbf{j} = 0 \)
- \( \mathbf{j} \cdot \mathbf{k} = 0 \)
- \( \mathbf{k} \cdot \mathbf{i} = 0 \)

2. Therefore, we can conclude:
\[ \mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} = 0 \]

**Answer for (iv):** \( 0 \)

### Summary of Answers:
- (i) \( 3 \)
- (ii) \( 0 \)
- (iii) \( 1 \)
- (iv) \( 0 \)