Let `veca , vecb , vecc` be three unit vectors such that ` |veca + vecb +vecc| =1 and veca bot vecb , " if " vecc` makes angles ` delta beta " with " veca, vecb` respectively, then ` cos delta + cos beta ` is equal to

Options

A. `(-3)/(2)`
B. `(3)/(2)`
C. 1
D. -1

Answer

To solve the problem, we need to analyze the given conditions and derive the required expression step by step.

### Given:
- Let \(\vec{a}, \vec{b}, \vec{c}\) be three unit vectors.
- \(|\vec{a} + \vec{b} + \vec{c}| = 1\)
- \(\vec{a} \perp \vec{b}\) (i.e., \(\vec{a}\) is perpendicular to \(\vec{b}\))
- \(\vec{c}\) makes angles \(\delta\) and \(\beta\) with \(\vec{a}\) and \(\vec{b}\) respectively.

### Step 1: Express the vectors
Assume:
- \(\vec{a} = \hat{i}\)
- \(\vec{b} = \hat{j}\)
- \(\vec{c} = -\hat{i}\)

### Step 2: Calculate \(|\vec{a} + \vec{b} + \vec{c}|\)
Now, we compute:
\[ \vec{a} + \vec{b} + \vec{c} = \hat{i} + \hat{j} - \hat{i} = \hat{j} \]
Thus,
\[ |\vec{a} + \vec{b} + \vec{c}| = |\hat{j}| = 1 \]
This satisfies the condition given in the problem.

### Step 3: Calculate \(\cos \delta\)
Since \(\vec{c}\) makes an angle \(\delta\) with \(\vec{a}\):
\[ \vec{a} \cdot \vec{c} = |\vec{a}| |\vec{c}| \cos \delta \]
Substituting the values:
\[ \hat{i} \cdot (-\hat{i}) = 1 \cdot 1 \cdot \cos \delta \]
This simplifies to:
\[ -1 = \cos \delta \]
Thus,
\[ \cos \delta = -1 \]

### Step 4: Calculate \(\cos \beta\)
Now, \(\vec{c}\) makes an angle \(\beta\) with \(\vec{b}\):
\[ \vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos \beta \]
Substituting the values:
\[ \hat{j} \cdot (-\hat{i}) = 1 \cdot 1 \cdot \cos \beta \]
This simplifies to:
\[ 0 = \cos \beta \]
Thus,
\[ \cos \beta = 0 \]

### Step 5: Calculate \(\cos \delta + \cos \beta\)
Now we combine the results:
\[ \cos \delta + \cos \beta = -1 + 0 = -1 \]

### Final Answer
\[ \cos \delta + \cos \beta = -1 \]

---