If vectors `4vecp+vecq, 2 vecp-3vecq and 5vecp-3vecq, 5vecp+3vecq` are a pair of mutually perpendicular vectors and if the angle between `vecp and vecq` is `theta`, then the value of `sin^(2)theta` is equal to

Options

A. `(3)/(5)`
B. `(9)/(25)`
C. `(1)/(2500)`
D. `(2499)/(2500)`

Answer

To solve the problem, we need to find the value of \( \sin^2 \theta \) given that the vectors \( 4\vec{p} + \vec{q} \), \( 2\vec{p} - 3\vec{q} \), \( 5\vec{p} - 3\vec{q} \), and \( 5\vec{p} + 3\vec{q} \) are mutually perpendicular.

### Step 1: Set up the equations for perpendicular vectors
Since the vectors are mutually perpendicular, the dot product of any two vectors must equal zero. We can start with the first pair:

\[ (4\vec{p} + \vec{q}) \cdot (2\vec{p} - 3\vec{q}) = 0 \]

### Step 2: Calculate the dot product
Expanding the dot product:

\[ 4\vec{p} \cdot 2\vec{p} + 4\vec{p} \cdot (-3\vec{q}) + \vec{q} \cdot 2\vec{p} + \vec{q} \cdot (-3\vec{q}) = 0 \]

This simplifies to:

\[ 8\vec{p} \cdot \vec{p} - 12\vec{p} \cdot \vec{q} + 2\vec{q} \cdot \vec{p} - 3\vec{q} \cdot \vec{q} = 0 \]

Using \( \vec{p} \cdot \vec{q} = \vec{q} \cdot \vec{p} \), we can combine terms:

\[ 8|\vec{p}|^2 - 10\vec{p} \cdot \vec{q} - 3|\vec{q}|^2 = 0 \tag{1} \]

### Step 3: Set up the second equation for the other pair
Now consider the second pair of vectors:

\[ (5\vec{p} + 3\vec{q}) \cdot (5\vec{p} - 3\vec{q}) = 0 \]

Expanding this:

\[ (5\vec{p}) \cdot (5\vec{p}) + (5\vec{p}) \cdot (-3\vec{q}) + (3\vec{q}) \cdot (5\vec{p}) + (3\vec{q}) \cdot (-3\vec{q}) = 0 \]

This simplifies to:

\[ 25|\vec{p}|^2 - 9|\vec{q}|^2 = 0 \tag{2} \]

### Step 4: Solve the equations
From equation (2):

\[ 25|\vec{p}|^2 = 9|\vec{q}|^2 \implies |\vec{p}|^2 = \frac{9}{25}|\vec{q}|^2 \]

Substituting this into equation (1):

\[ 8\left(\frac{9}{25}|\vec{q}|^2\right) - 10\vec{p} \cdot \vec{q} - 3|\vec{q}|^2 = 0 \]

This simplifies to:

\[ \frac{72}{25}|\vec{q}|^2 - 10\vec{p} \cdot \vec{q} - 3|\vec{q}|^2 = 0 \]

Combining the terms gives:

\[ \left(\frac{72}{25} - 3\right)|\vec{q}|^2 = 10\vec{p} \cdot \vec{q} \]

Converting \( 3 \) to a fraction:

\[ \frac{72}{25} - \frac{75}{25} = -\frac{3}{25} \implies -\frac{3}{25}|\vec{q}|^2 = 10\vec{p} \cdot \vec{q} \]

Thus,

\[ \vec{p} \cdot \vec{q} = -\frac{3}{250}|\vec{q}|^2 \]

### Step 5: Find \( \cos \theta \)
Using the relationship \( \cos \theta = \frac{\vec{p} \cdot \vec{q}}{|\vec{p}||\vec{q}|} \):

Substituting the value of \( \vec{p} \cdot \vec{q} \):

\[ \cos \theta = \frac{-\frac{3}{250}|\vec{q}|^2}{|\vec{p}||\vec{q}|} \]

Using \( |\vec{p}| = \frac{3}{5}|\vec{q}| \):

\[ \cos \theta = \frac{-\frac{3}{250}|\vec{q}|^2}{\frac{3}{5}|\vec{q}||\vec{q}|} = -\frac{1}{50} \]

### Step 6: Calculate \( \sin^2 \theta \)
Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \):

\[ \cos^2 \theta = \left(-\frac{1}{50}\right)^2 = \frac{1}{2500} \]

Thus,

\[ \sin^2 \theta = 1 - \frac{1}{2500} = \frac{2499}{2500} \]

### Final Answer
The value of \( \sin^2 \theta \) is:

\[ \sin^2 \theta = \frac{2499}{2500} \]