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 dot product equations
Since the vectors are mutually perpendicular, we can set up the following equations based on the dot product being zero:

1. \( (4\vec{p} + \vec{q}) \cdot (2\vec{p} - 3\vec{q}) = 0 \)
2. \( (5\vec{p} - 3\vec{q}) \cdot (5\vec{p} + 3\vec{q}) = 0 \)

### Step 2: Calculate the first dot product
Calculating the first dot product:

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

This simplifies to:

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

Combining like terms gives:

\[ 8|\vec{p}|^2 - 10 \vec{p} \cdot \vec{q} - 3|\vec{q}|^2 = 0 \quad \text{(Equation 1)} \]

### Step 3: Calculate the second dot product
Now calculating the second dot product:

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

This simplifies to:

\[ 25|\vec{p}|^2 - 9|\vec{q}|^2 = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
From Equation 2, we can express \( |\vec{p}|^2 \) in terms of \( |\vec{q}|^2 \):

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

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 terms gives:

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

Calculating \( \frac{72}{25} - 3 = \frac{72 - 75}{25} = \frac{-3}{25} \):

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

### Step 5: Find \( \cos \theta \)
Now we can express \( \vec{p} \cdot \vec{q} \):

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

Using the cosine formula:

\[ \cos \theta = \frac{\vec{p} \cdot \vec{q}}{|\vec{p}||\vec{q}|} \]

Substituting \( |\vec{p}| = \sqrt{\frac{9}{25}}|\vec{q}| = \frac{3}{5}|\vec{q}| \):

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

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

\[ \sin^2 \theta = 1 - \left(-\frac{1}{50}\right)^2 = 1 - \frac{1}{2500} = \frac{2499}{2500} \]

Thus, the final answer is:

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