If `| vec(a) | =3` and `|vec(b) |=4`, then the value of `lambda` for which `vec(a) + lambda vec(b) ` and `vec(a) - lambda vec(b)` are perpendicular 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 vectors \( \vec{a} + \lambda \vec{b} \) and \( \vec{a} - \lambda \vec{b} \) are perpendicular.

### Step-by-Step Solution:

1. **Understanding the Condition for Perpendicular Vectors**:
Two vectors \( \vec{u} \) and \( \vec{v} \) are perpendicular if their dot product is zero:
\[ \vec{u} \cdot \vec{v} = 0 \]

2. **Setting Up the Vectors**:
Let \( \vec{u} = \vec{a} + \lambda \vec{b} \) and \( \vec{v} = \vec{a} - \lambda \vec{b} \).

3. **Calculating the Dot Product**:
We need to calculate the dot product \( \vec{u} \cdot \vec{v} \):
\[ \vec{u} \cdot \vec{v} = (\vec{a} + \lambda \vec{b}) \cdot (\vec{a} - \lambda \vec{b}) \]

4. **Expanding the Dot Product**:
Using the distributive property of the dot product, we have:
\[ \vec{u} \cdot \vec{v} = \vec{a} \cdot \vec{a} - \lambda \vec{a} \cdot \vec{b} + \lambda \vec{b} \cdot \vec{a} - \lambda^2 \vec{b} \cdot \vec{b} \]
Since \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \), this simplifies to:
\[ \vec{u} \cdot \vec{v} = |\vec{a}|^2 - \lambda^2 |\vec{b}|^2 \]

5. **Substituting the Magnitudes**:
We know that \( |\vec{a}| = 3 \) and \( |\vec{b}| = 4 \). Therefore:
\[ |\vec{a}|^2 = 3^2 = 9 \quad \text{and} \quad |\vec{b}|^2 = 4^2 = 16 \]
Substituting these values gives:
\[ \vec{u} \cdot \vec{v} = 9 - \lambda^2 \cdot 16 \]

6. **Setting the Dot Product to Zero**:
For the vectors to be perpendicular, we set the dot product to zero:
\[ 9 - 16\lambda^2 = 0 \]

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

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

9. **Choosing the Positive Value**:
Since the options provided are positive, we take:
\[ \lambda = \frac{3}{4} \]

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