If `|bar(a)| = 3,|bar(b)| = 4`, then the value of `lambda` for which `bar(a) + lambda bar(b)` is perpendicular to `bar(a) - lambda bar(b)` is…..

Options

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

Answer

To find the value of \( \lambda \) for which the vectors \( \bar{a} + \lambda \bar{b} \) and \( \bar{a} - \lambda \bar{b} \) are perpendicular, we can follow these steps:

### Step 1: Understand the condition for perpendicular vectors
Two vectors \( \bar{u} \) and \( \bar{v} \) are perpendicular if their dot product is zero:
\[ \bar{u} \cdot \bar{v} = 0 \]

### Step 2: Set up the dot product
We need to compute the dot product of \( \bar{a} + \lambda \bar{b} \) and \( \bar{a} - \lambda \bar{b} \):
\[ (\bar{a} + \lambda \bar{b}) \cdot (\bar{a} - \lambda \bar{b}) = 0 \]

### Step 3: Expand the dot product
Using the distributive property of the dot product:
\[ \bar{a} \cdot \bar{a} - \lambda \bar{a} \cdot \bar{b} + \lambda \bar{b} \cdot \bar{a} - \lambda^2 \bar{b} \cdot \bar{b} = 0 \]
Since \( \bar{a} \cdot \bar{b} = \bar{b} \cdot \bar{a} \), we can simplify this to:
\[ \bar{a} \cdot \bar{a} - \lambda^2 \bar{b} \cdot \bar{b} = 0 \]

### Step 4: Substitute the magnitudes
Given that \( |\bar{a}| = 3 \) and \( |\bar{b}| = 4 \):
\[ |\bar{a}|^2 = 3^2 = 9 \quad \text{and} \quad |\bar{b}|^2 = 4^2 = 16 \]
Substituting these values into the equation:
\[ 9 - \lambda^2 \cdot 16 = 0 \]

### Step 5: Solve for \( \lambda^2 \)
Rearranging the equation gives:
\[ \lambda^2 \cdot 16 = 9 \]
\[ \lambda^2 = \frac{9}{16} \]

### Step 6: Take the square root
Taking the square root of both sides:
\[ \lambda = \pm \frac{3}{4} \]

### Conclusion
The values of \( \lambda \) for which \( \bar{a} + \lambda \bar{b} \) is perpendicular to \( \bar{a} - \lambda \bar{b} \) are:
\[ \lambda = \frac{3}{4} \quad \text{and} \quad \lambda = -\frac{3}{4} \]