Given that `veca,vecb,vecp,vecq` are four vectors such that `veca + vecb= mu vecp, vecb.vecq=0 and |vecb|^(2) = 1 ` where `mu` is a sclar. Then `|(veca.vecq) vecp-(vecp.vecq)veca|` is equal to

(a)`2|vecpvecq|` (b)`(1//2)|vecp.vecq|` (c)`|vecpxxvecq|` (d)`|vecp.vecq|`

Options

A. `2|vecpvecq|`
B. `(1//2)|vecp.vecq|`
C. `|vecpxxvecq|`
D. `|vecp.vecq|`

Explanation

To solve the problem, we need to find the value of \( |(\vec{a} \cdot \vec{q}) \vec{p} - (\vec{p} \cdot \vec{q}) \vec{a}| \) given the conditions on the vectors. Let's break it down step by step.

### Step 1: Start with the given equations
We have the following equations:
1. \( \vec{a} + \vec{b} = \mu \vec{p} \)
2. \( \vec{b} \cdot \vec{q} = 0 \)
3. \( |\vec{b}|^2 = 1 \)

### Step 2: Take the dot product with \( \vec{q} \)
Taking the dot product of the first equation with \( \vec{q} \):
\[ \vec{a} \cdot \vec{q} + \vec{b} \cdot \vec{q} = \mu (\vec{p} \cdot \vec{q}) \]
Since \( \vec{b} \cdot \vec{q} = 0 \), we can simplify this to:
\[ \vec{a} \cdot \vec{q} = \mu (\vec{p} \cdot \vec{q}) \]

### Step 3: Express \( \vec{b} \) in terms of \( \vec{a} \) and \( \vec{p} \)
From the first equation, we can express \( \vec{b} \) as:
\[ \vec{b} = \mu \vec{p} - \vec{a} \]

### Step 4: Substitute \( \vec{b} \) into the expression
Now we need to evaluate the expression \( |(\vec{a} \cdot \vec{q}) \vec{p} - (\vec{p} \cdot \vec{q}) \vec{a}| \):
Substituting \( \vec{a} \cdot \vec{q} = \mu (\vec{p} \cdot \vec{q}) \):
\[ |(\mu (\vec{p} \cdot \vec{q})) \vec{p} - (\vec{p} \cdot \vec{q}) \vec{a}| \]

### Step 5: Factor out \( \vec{p} \cdot \vec{q} \)
Factoring out \( \vec{p} \cdot \vec{q} \):
\[ |(\vec{p} \cdot \vec{q}) (\mu \vec{p} - \vec{a})| \]

### Step 6: Use the magnitude of \( \vec{b} \)
From the third equation, we know \( |\vec{b}|^2 = 1 \), hence \( |\vec{b}| = 1 \):
\[ |\mu \vec{p} - \vec{a}| = |\vec{b}| \]
Thus, we have:
\[ |\mu \vec{p} - \vec{a}| = 1 \]

### Step 7: Final expression
Now substituting back, we find:
\[ |(\vec{p} \cdot \vec{q})| \cdot 1 = |\vec{p} \cdot \vec{q}| \]

### Conclusion
Thus, the final result is:
\[ |(\vec{a} \cdot \vec{q}) \vec{p} - (\vec{p} \cdot \vec{q}) \vec{a}| = |\vec{p} \cdot \vec{q}| \]
The answer is option **(d)** \( |\vec{p} \cdot \vec{q}| \).

---