Let `veca` and `vecb` be two vectors of equal magnitude 5 units. Let `vecp,vecq` be vectors such that `vecp=veca-vecb` and `vecq=veca+vecb`. If `|vecp xx vecq|=2{lambda-(veca.vecb)^(2)}^(1/2)`, then value of `lambda` is

Options

A. 25
B. 125
C. 625
D. none of these

Explanation

To solve the problem step by step, we start with the given vectors and their properties.

### Step 1: Define the vectors
Let \(\vec{a}\) and \(\vec{b}\) be two vectors of equal magnitude, which is given as 5 units. Therefore, we have:
\[ |\vec{a}| = |\vec{b}| = 5 \]

### Step 2: Define \(\vec{p}\) and \(\vec{q}\)
We are given:
\[ \vec{p} = \vec{a} - \vec{b} \]
\[ \vec{q} = \vec{a} + \vec{b} \]

### Step 3: Find \(|\vec{p} \times \vec{q}|\)
Using the property of cross products, we can express \(\vec{p} \times \vec{q}\) as follows:
\[ \vec{p} \times \vec{q} = (\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) \]
Using the distributive property of the cross product:
\[ \vec{p} \times \vec{q} = \vec{a} \times \vec{a} + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} - \vec{b} \times \vec{b} \]
Since the cross product of any vector with itself is zero:
\[ \vec{a} \times \vec{a} = 0 \quad \text{and} \quad \vec{b} \times \vec{b} = 0 \]
Thus, we have:
\[ \vec{p} \times \vec{q} = \vec{a} \times \vec{b} - \vec{b} \times \vec{a} \]
Using the property \(\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})\), we can simplify this to:
\[ \vec{p} \times \vec{q} = 2(\vec{a} \times \vec{b}) \]

### Step 4: Find the magnitude of \(|\vec{p} \times \vec{q}|\)
The magnitude of the cross product is given by:
\[ |\vec{p} \times \vec{q}| = |2(\vec{a} \times \vec{b})| = 2|\vec{a} \times \vec{b}| \]

### Step 5: Calculate \(|\vec{a} \times \vec{b}|\)
The magnitude of the cross product \(|\vec{a} \times \vec{b}|\) can be expressed using the formula:
\[ |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta \]
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\). Since both vectors have a magnitude of 5:
\[ |\vec{a} \times \vec{b}| = 5 \cdot 5 \cdot \sin\theta = 25\sin\theta \]
Thus:
\[ |\vec{p} \times \vec{q}| = 2 \cdot 25\sin\theta = 50\sin\theta \]

### Step 6: Use the given equation
We are given:
\[ |\vec{p} \times \vec{q}| = 2\sqrt{\lambda - (\vec{a} \cdot \vec{b})^2} \]
Equating the two expressions for \(|\vec{p} \times \vec{q}|\):
\[ 50\sin\theta = 2\sqrt{\lambda - (\vec{a} \cdot \vec{b})^2} \]

### Step 7: Square both sides
Squaring both sides gives:
\[ (50\sin\theta)^2 = 4(\lambda - (\vec{a} \cdot \vec{b})^2) \]
This simplifies to:
\[ 2500\sin^2\theta = 4\lambda - 4(\vec{a} \cdot \vec{b})^2 \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ 4\lambda = 2500\sin^2\theta + 4(\vec{a} \cdot \vec{b})^2 \]
\[ \lambda = 625\sin^2\theta + (\vec{a} \cdot \vec{b})^2 \]

### Step 9: Find the value of \(\lambda\)
Since \(\vec{a}\) and \(\vec{b}\) are both 5 units in magnitude, the maximum value of \(\vec{a} \cdot \vec{b}\) occurs when \(\theta = 0\) (i.e., when they are in the same direction), giving:
\[ \vec{a} \cdot \vec{b} = 5 \cdot 5 = 25 \]
Thus:
\[ \lambda = 625 + 25 = 650 \]

### Final Answer
The value of \(\lambda\) is:
\[ \lambda = 625 \]