If `|veca| =3,|vecb| = 4` , then a value of `lamda` for which `veca + lamda vecb` is perpendicular to `veca - lamda vecb` is :

Options

A. `9/16`
B. `3/4`
C. `3/2`
D. `4/2`

Answer

To find the value of \( \lambda \) for which \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{a} - \lambda \vec{b} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Perpendicular Vectors**:
Two vectors \( \vec{u} \) and \( \vec{v} \) are perpendicular if their dot product is zero, i.e., \( \vec{u} \cdot \vec{v} = 0 \).

2. **Setting Up the Equation**:
We need to find \( \lambda \) such that:
\[ (\vec{a} + \lambda \vec{b}) \cdot (\vec{a} - \lambda \vec{b}) = 0 \]

3. **Expanding the Dot Product**:
Using the distributive property of the dot product:
\[ \vec{a} \cdot \vec{a} - \lambda \vec{a} \cdot \vec{b} + \lambda \vec{b} \cdot \vec{a} - \lambda^2 \vec{b} \cdot \vec{b} = 0 \]
Since \( \vec{a} \cdot \vec{b} \) is a scalar, we can simplify this to:
\[ |\vec{a}|^2 - \lambda^2 |\vec{b}|^2 = 0 \]

4. **Substituting the Magnitudes**:
We know that \( |\vec{a}| = 3 \) and \( |\vec{b}| = 4 \). Thus:
\[ 3^2 - \lambda^2 \cdot 4^2 = 0 \]
This simplifies to:
\[ 9 - 16\lambda^2 = 0 \]

5. **Solving for \( \lambda^2 \)**:
Rearranging gives:
\[ 16\lambda^2 = 9 \]
Dividing both sides by 16:
\[ \lambda^2 = \frac{9}{16} \]

6. **Taking the Square Root**:
Taking the square root of both sides gives:
\[ \lambda = \pm \frac{3}{4} \]

### Final Answer:
The values of \( \lambda \) for which \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{a} - \lambda \vec{b} \) are \( \lambda = \frac{3}{4} \) and \( \lambda = -\frac{3}{4} \).