What is the angle between `(vec(P)+vec(Q))` and `(vec(P)xxvecQ)?`

Options

A. Zero
B. `(pi)/(2)`
C. `(pi)/(4)`
D. `pi`

Explanation

To find the angle between the vectors \(\vec{P} + \vec{Q}\) and \(\vec{P} \times \vec{Q}\), we can follow these steps:

### Step 1: Understand the Properties of Vectors
- The vector \(\vec{P} + \vec{Q}\) represents the resultant of the two vectors \(\vec{P}\) and \(\vec{Q}\).
- The vector \(\vec{P} \times \vec{Q}\) represents the cross product of \(\vec{P}\) and \(\vec{Q}\), which is a vector that is perpendicular to both \(\vec{P}\) and \(\vec{Q}\).

### Step 2: Analyze the Geometric Arrangement
- The vector \(\vec{P} + \vec{Q}\) lies in the plane formed by \(\vec{P}\) and \(\vec{Q}\).
- The vector \(\vec{P} \times \vec{Q}\) is perpendicular to this plane.

### Step 3: Determine the Relationship Between the Vectors
- Since \(\vec{P} + \vec{Q}\) lies in the same plane as \(\vec{P}\) and \(\vec{Q}\), and \(\vec{P} \times \vec{Q}\) is perpendicular to this plane, it follows that \(\vec{P} + \vec{Q}\) is also perpendicular to \(\vec{P} \times \vec{Q}\).

### Step 4: Conclusion
- The angle between two vectors is defined as \(90^\circ\) when they are perpendicular to each other.
- Therefore, the angle between \(\vec{P} + \vec{Q}\) and \(\vec{P} \times \vec{Q}\) is \(90^\circ\).

### Final Answer
The angle between \((\vec{P} + \vec{Q})\) and \((\vec{P} \times \vec{Q})\) is \(90^\circ\).

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