The corner P of the square OPQR is folded up so that the plane OPQ is perpendicular to the plane OQR, find the angle between OP and QR.

Answer

To solve the problem of finding the angle between OP and QR after folding the corner P of the square OPQR, we can follow these steps:

### Step 1: Define the Position Vectors
We start by defining the position vectors of the points in the square OPQR. Let's assume the square has a side length of \( A \).

- \( O \) (origin) is at \( (0, 0, 0) \)
- \( P \) is at \( (A, 0, 0) \)
- \( Q \) is at \( (A, A, 0) \)
- \( R \) is at \( (0, A, 0) \)

### Step 2: Fold Point P Upwards
When we fold point \( P \) upwards, it moves to a new position \( P' \) such that the plane \( OPQ \) is perpendicular to the plane \( OQR \).

Let’s denote the new position of \( P \) as \( P' \). The coordinates of \( P' \) can be expressed as:
- \( P' = (A, 0, h) \) where \( h \) is the height to which \( P \) is folded.

### Step 3: Find the Vectors OP and QR
Next, we need to find the vectors \( \overrightarrow{OP} \) and \( \overrightarrow{QR} \).

- The vector \( \overrightarrow{OP} \) is given by:
\[ \overrightarrow{OP} = P - O = (A, 0, 0) - (0, 0, 0) = (A, 0, 0) \]

- The vector \( \overrightarrow{QR} \) is given by:
\[ \overrightarrow{QR} = R - Q = (0, A, 0) - (A, A, 0) = (-A, 0, 0) \]

### Step 4: Calculate the Dot Product
To find the angle \( \theta \) between the vectors \( \overrightarrow{OP} \) and \( \overrightarrow{QR} \), we use the dot product formula:
\[ \overrightarrow{OP} \cdot \overrightarrow{QR} = |\overrightarrow{OP}| |\overrightarrow{QR}| \cos \theta \]

Calculating the dot product:
\[ \overrightarrow{OP} \cdot \overrightarrow{QR} = (A, 0, 0) \cdot (-A, 0, 0) = -A^2 \]

Calculating the magnitudes:
\[ |\overrightarrow{OP}| = A \quad \text{and} \quad |\overrightarrow{QR}| = A \]

### Step 5: Substitute into the Dot Product Formula
Substituting into the dot product formula:
\[ -A^2 = A \cdot A \cdot \cos \theta \]
\[ -A^2 = A^2 \cos \theta \]

### Step 6: Solve for Cosine
Dividing both sides by \( A^2 \) (assuming \( A \neq 0 \)):
\[ \cos \theta = -1 \]

### Step 7: Find the Angle
The angle \( \theta \) for which \( \cos \theta = -1 \) is:
\[ \theta = 180^\circ \quad \text{or} \quad \theta = \pi \text{ radians} \]

### Conclusion
Thus, the angle between \( OP \) and \( QR \) after folding is \( 180^\circ \).

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