If `vecP xx vecQ=vecQ xx vecP`, the angle between `vecP and vecQ` is `theta(0^(@) lt theta lt 360^(@))`. The value of '`theta`' will be __________`""^(@)`.

Answer

To solve the problem, we start with the equation given:

\[ \vec{P} \times \vec{Q} = \vec{Q} \times \vec{P} \]

### Step 1: Understanding the Cross Product
The cross product of two vectors \(\vec{A}\) and \(\vec{B}\) is defined as:

\[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \hat{n} \]

where \(\theta\) is the angle between the vectors, and \(\hat{n}\) is the unit vector perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\).

### Step 2: Analyzing the Given Equation
From the equation \(\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}\), we can rewrite the right side using the property of the cross product:

\[ \vec{Q} \times \vec{P} = -(\vec{P} \times \vec{Q}) \]

Substituting this into our equation gives:

\[ \vec{P} \times \vec{Q} = -(\vec{P} \times \vec{Q}) \]

### Step 3: Simplifying the Equation
Adding \(\vec{P} \times \vec{Q}\) to both sides results in:

\[ \vec{P} \times \vec{Q} + \vec{P} \times \vec{Q} = 0 \]

This simplifies to:

\[ 2(\vec{P} \times \vec{Q}) = 0 \]

### Step 4: Conclusion from the Cross Product
Since the product is zero, we have:

\[ \vec{P} \times \vec{Q} = 0 \]

The condition for the cross product to be zero is that either one of the vectors is zero or the sine of the angle between them is zero:

\[ |\vec{P}| |\vec{Q}| \sin(\theta) = 0 \]

### Step 5: Finding the Angle
Since we are not considering the case where either vector is zero, we focus on:

\[ \sin(\theta) = 0 \]

The sine function is zero at angles:

\[ \theta = 0^\circ, 180^\circ, 360^\circ \]

However, since the problem states that \(0^\circ < \theta < 360^\circ\), the only valid solution is:

\[ \theta = 180^\circ \]

Thus, the value of \(\theta\) is:

\[ \boxed{180^\circ} \]