If for any two vectors `veca and vecb.` `(veca + vecb )^(2) + (veca - vecb)^(2) =lamda[(veca)^(2)+ (vecb)^(2)]` then write the value of `lamda.`

Answer

To solve the problem, we need to analyze the given equation step by step. The equation is:

\[ (\vec{a} + \vec{b})^2 + (\vec{a} - \vec{b})^2 = \lambda [\vec{a}^2 + \vec{b}^2] \]

### Step 1: Expand the Left-Hand Side (LHS)

We start by expanding the left-hand side of the equation:

\[ (\vec{a} + \vec{b})^2 = \vec{a} \cdot \vec{a} + 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \|\vec{a}\|^2 + 2\vec{a} \cdot \vec{b} + \|\vec{b}\|^2 \]

\[ (\vec{a} - \vec{b})^2 = \vec{a} \cdot \vec{a} - 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \|\vec{a}\|^2 - 2\vec{a} \cdot \vec{b} + \|\vec{b}\|^2 \]

Now, we add these two expansions together:

\[ (\vec{a} + \vec{b})^2 + (\vec{a} - \vec{b})^2 = \left( \|\vec{a}\|^2 + 2\vec{a} \cdot \vec{b} + \|\vec{b}\|^2 \right) + \left( \|\vec{a}\|^2 - 2\vec{a} \cdot \vec{b} + \|\vec{b}\|^2 \right) \]

### Step 2: Simplify the LHS

Combining the terms, we get:

\[ = 2\|\vec{a}\|^2 + 2\|\vec{b}\|^2 \]

### Step 3: Rewrite the LHS

Thus, we can rewrite the left-hand side as:

\[ 2(\|\vec{a}\|^2 + \|\vec{b}\|^2) \]

### Step 4: Set the LHS Equal to the RHS

Now, we equate the left-hand side to the right-hand side of the original equation:

\[ 2(\|\vec{a}\|^2 + \|\vec{b}\|^2) = \lambda (\|\vec{a}\|^2 + \|\vec{b}\|^2) \]

### Step 5: Solve for \(\lambda\)

Assuming \(\|\vec{a}\|^2 + \|\vec{b}\|^2 \neq 0\), we can divide both sides by \((\|\vec{a}\|^2 + \|\vec{b}\|^2)\):

\[ 2 = \lambda \]

Thus, we find:

\[ \lambda = 2 \]

### Final Answer

The value of \(\lambda\) is:

\[ \lambda = 2 \]

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