Let `bara and barb` be two vectors of equal magnitude 5units. Let `barp,q` be vectors such that `barp=bara+barb and barq=bara-barb`. IF `|barp times barq|=2(lamda-(bara.barb)^2}^(1/2)` then value of `lamda` is

Options

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

Answer

To solve the problem step by step, let's define the vectors and their properties clearly.

### Step 1: Define the vectors
Let \( \vec{a} \) and \( \vec{b} \) be two vectors with equal magnitudes:
\[ |\vec{a}| = |\vec{b}| = 5 \text{ units} \]

### Step 2: Define the new vectors
We define two new vectors:
\[ \vec{p} = \vec{a} + \vec{b} \]
\[ \vec{q} = \vec{a} - \vec{b} \]

### Step 3: Calculate the magnitudes of \( \vec{p} \) and \( \vec{q} \)
To find the magnitudes of \( \vec{p} \) and \( \vec{q} \), we use the formula for the magnitude of a vector:
\[ |\vec{p}| = |\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b})} \]
Substituting the values:
\[ |\vec{p}| = \sqrt{5^2 + 5^2 + 2(\vec{a} \cdot \vec{b})} = \sqrt{50 + 2(\vec{a} \cdot \vec{b})} \]

Similarly, for \( \vec{q} \):
\[ |\vec{q}| = |\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - 2(\vec{a} \cdot \vec{b})} \]
Substituting the values:
\[ |\vec{q}| = \sqrt{5^2 + 5^2 - 2(\vec{a} \cdot \vec{b})} = \sqrt{50 - 2(\vec{a} \cdot \vec{b})} \]

### Step 4: Calculate \( |\vec{p} \times \vec{q}| \)
The magnitude of the cross product of two vectors is given by:
\[ |\vec{p} \times \vec{q}| = |\vec{p}| |\vec{q}| \sin \theta \]
where \( \theta \) is the angle between \( \vec{p} \) and \( \vec{q} \).

### Step 5: Find \( \vec{p} \cdot \vec{q} \)
We can find the dot product:
\[ \vec{p} \cdot \vec{q} = (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{b} = |\vec{a}|^2 - |\vec{b}|^2 = 25 - 25 + 2(\vec{a} \cdot \vec{b}) = 2(\vec{a} \cdot \vec{b}) \]

### Step 6: Use the relationship given in the problem
We know from the problem statement:
\[ |\vec{p} \times \vec{q}| = 2\left(\lambda - (\vec{a} \cdot \vec{b})^2\right)^{1/2} \]

### Step 7: Substitute values into the equation
From the previous steps:
\[ |\vec{p}| = \sqrt{50 + 2(\vec{a} \cdot \vec{b})} \]
\[ |\vec{q}| = \sqrt{50 - 2(\vec{a} \cdot \vec{b})} \]
Thus:
\[ |\vec{p} \times \vec{q}| = \sqrt{(50 + 2(\vec{a} \cdot \vec{b}))(50 - 2(\vec{a} \cdot \vec{b}))} \cdot \sin \theta \]
This simplifies to:
\[ |\vec{p} \times \vec{q}| = \sqrt{2500 - 4(\vec{a} \cdot \vec{b})^2} \cdot \sin \theta \]

### Step 8: Set the two expressions equal
Setting the two expressions for \( |\vec{p} \times \vec{q}| \) equal gives:
\[ \sqrt{2500 - 4(\vec{a} \cdot \vec{b})^2} \cdot \sin \theta = 2\left(\lambda - (\vec{a} \cdot \vec{b})^2\right)^{1/2} \]

### Step 9: Solve for \( \lambda \)
To find \( \lambda \), we need to isolate it. From the equality, we can square both sides and simplify:
\[ 2500 - 4(\vec{a} \cdot \vec{b})^2 = 4\left(\lambda - (\vec{a} \cdot \vec{b})^2\right) \]
Expanding and rearranging gives:
\[ 2500 = 4\lambda - 4(\vec{a} \cdot \vec{b})^2 + 4(\vec{a} \cdot \vec{b})^2 \]
Thus:
\[ 2500 = 4\lambda \]
Finally, solving for \( \lambda \):
\[ \lambda = \frac{2500}{4} = 625 \]

### Final Answer
The value of \( \lambda \) is \( 625 \).