Let `a =i+j+pk and b=i+j+k, |a+b| =|a| +|b|` , holds for

Options

A. all real p
B. no real p
C. one real p
D. two real p

Answer

To solve the problem, we need to analyze the vectors \( \mathbf{a} \) and \( \mathbf{b} \) given by:

\[ \mathbf{a} = \mathbf{i} + \mathbf{j} + p\mathbf{k} \]
\[ \mathbf{b} = \mathbf{i} + \mathbf{j} + \mathbf{k} \]

We are tasked with finding the values of \( p \) for which the equation \( |\mathbf{a} + \mathbf{b}| = |\mathbf{a}| + |\mathbf{b}| \) holds true.

### Step 1: Calculate \( \mathbf{a} + \mathbf{b} \)

First, we need to find \( \mathbf{a} + \mathbf{b} \):

\[ \mathbf{a} + \mathbf{b} = (\mathbf{i} + \mathbf{j} + p\mathbf{k}) + (\mathbf{i} + \mathbf{j} + \mathbf{k}) = 2\mathbf{i} + 2\mathbf{j} + (p + 1)\mathbf{k} \]

### Step 2: Calculate \( |\mathbf{a} + \mathbf{b}| \)

Now, we calculate the magnitude \( |\mathbf{a} + \mathbf{b}| \):

\[ |\mathbf{a} + \mathbf{b}| = \sqrt{(2)^2 + (2)^2 + (p + 1)^2} = \sqrt{4 + 4 + (p + 1)^2} = \sqrt{8 + (p + 1)^2} \]

### Step 3: Calculate \( |\mathbf{a}| \) and \( |\mathbf{b}| \)

Next, we calculate the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \):

\[ |\mathbf{a}| = \sqrt{(1)^2 + (1)^2 + (p)^2} = \sqrt{1 + 1 + p^2} = \sqrt{2 + p^2} \]

\[ |\mathbf{b}| = \sqrt{(1)^2 + (1)^2 + (1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \]

### Step 4: Set up the equation

Now we set up the equation based on the condition given:

\[ |\mathbf{a} + \mathbf{b}| = |\mathbf{a}| + |\mathbf{b}| \]

Substituting the magnitudes we calculated:

\[ \sqrt{8 + (p + 1)^2} = \sqrt{2 + p^2} + \sqrt{3} \]

### Step 5: Square both sides

To eliminate the square roots, we square both sides:

\[ 8 + (p + 1)^2 = (\sqrt{2 + p^2} + \sqrt{3})^2 \]

Expanding both sides:

\[ 8 + (p^2 + 2p + 1) = (2 + p^2) + 3 + 2\sqrt{3}\sqrt{2 + p^2} \]

This simplifies to:

\[ p^2 + 2p + 9 = p^2 + 5 + 2\sqrt{3}\sqrt{2 + p^2} \]

### Step 6: Rearranging the equation

Now, we can rearrange the equation:

\[ 2p + 4 = 2\sqrt{3}\sqrt{2 + p^2} \]

Dividing both sides by 2:

\[ p + 2 = \sqrt{3}\sqrt{2 + p^2} \]

### Step 7: Square again

Squaring both sides again gives:

\[ (p + 2)^2 = 3(2 + p^2) \]

Expanding both sides:

\[ p^2 + 4p + 4 = 6 + 3p^2 \]

### Step 8: Rearranging terms

Rearranging the equation leads to:

\[ 0 = 2p^2 - 4p + 2 \]

Dividing through by 2:

\[ 0 = p^2 - 2p + 1 \]

### Step 9: Factoring

Factoring gives:

\[ (p - 1)^2 = 0 \]

### Step 10: Solution

Thus, we find:

\[ p = 1 \]

### Conclusion

The only real value of \( p \) for which \( |\mathbf{a} + \mathbf{b}| = |\mathbf{a}| + |\mathbf{b}| \) holds is:

\[ \boxed{1} \]