If `|vec(a)| = 3, |vec(b)| = 4`, then the value 'lambda' for which `vec(a) + lambda vec(b)` is perpendicular to `vec(a) - lambda vec(b)`, is

Options

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

Answer

To solve the problem, we need to find the value of \( \lambda \) for which the vector \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{a} - \lambda \vec{b} \).

### Step-by-Step Solution:

1. **Understanding Perpendicular Vectors**:
Two vectors \( \vec{u} \) and \( \vec{v} \) are perpendicular if their dot product is zero. Therefore, we need to set up the equation:
\[ (\vec{a} + \lambda \vec{b}) \cdot (\vec{a} - \lambda \vec{b}) = 0 \]

2. **Expanding the Dot Product**:
Using the distributive property of the dot product, we expand the left-hand side:
\[ \vec{a} \cdot \vec{a} - \lambda \vec{a} \cdot \vec{b} + \lambda \vec{b} \cdot \vec{a} - \lambda^2 \vec{b} \cdot \vec{b} \]
This simplifies to:
\[ |\vec{a}|^2 - \lambda^2 |\vec{b}|^2 = 0 \]
(Note: \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \))

3. **Substituting Magnitudes**:
We know that \( |\vec{a}| = 3 \) and \( |\vec{b}| = 4 \). Therefore:
\[ |\vec{a}|^2 = 3^2 = 9 \]
\[ |\vec{b}|^2 = 4^2 = 16 \]

4. **Setting Up the Equation**:
Substitute the magnitudes into the equation:
\[ 9 - \lambda^2 \cdot 16 = 0 \]

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

6. **Finding \( \lambda \)**:
Taking the square root of both sides, we get:
\[ \lambda = \pm \frac{3}{4} \]

### Conclusion:
The value of \( \lambda \) for which \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{a} - \lambda \vec{b} \) is \( \frac{3}{4} \) or \( -\frac{3}{4} \).