If P(0, 1, 2), Q(4, -2, 1) and R(0, 0, 0) are three
points, then `anglePRQ` is

Options

A. `pi/4`
B. `pi/2`
C. `pi/3`
D. `pi/6`

Answer

To find the angle \( \angle PRQ \) formed by the points \( P(0, 1, 2) \), \( Q(4, -2, 1) \), and \( R(0, 0, 0) \), we will follow these steps:

### Step 1: Find the vectors \( \overrightarrow{PR} \) and \( \overrightarrow{QR} \)

1. **Calculate vector \( \overrightarrow{PR} \)**:
\[ \overrightarrow{PR} = R - P = (0, 0, 0) - (0, 1, 2) = (0 - 0, 0 - 1, 0 - 2) = (0, -1, -2) \]

2. **Calculate vector \( \overrightarrow{QR} \)**:
\[ \overrightarrow{QR} = R - Q = (0, 0, 0) - (4, -2, 1) = (0 - 4, 0 - (-2), 0 - 1) = (-4, 2, -1) \]

### Step 2: Identify the direction ratios

- The direction ratios of \( \overrightarrow{PR} \) are \( (0, -1, -2) \).
- The direction ratios of \( \overrightarrow{QR} \) are \( (-4, 2, -1) \).

### Step 3: Use the formula for the cosine of the angle between two vectors

The cosine of the angle \( \theta \) between two vectors can be calculated using the formula:
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}} \]
Where \( (l_1, m_1, n_1) \) are the direction ratios of \( \overrightarrow{PR} \) and \( (l_2, m_2, n_2) \) are the direction ratios of \( \overrightarrow{QR} \).

### Step 4: Substitute the values into the formula

- For \( \overrightarrow{PR} \): \( l_1 = 0, m_1 = -1, n_1 = -2 \)
- For \( \overrightarrow{QR} \): \( l_2 = -4, m_2 = 2, n_2 = -1 \)

Substituting these values into the formula:
\[ \cos \theta = \frac{(0)(-4) + (-1)(2) + (-2)(-1)}{\sqrt{0^2 + (-1)^2 + (-2)^2} \cdot \sqrt{(-4)^2 + 2^2 + (-1)^2}} \]

### Step 5: Simplify the numerator and denominator

1. **Numerator**:
\[ 0 - 2 + 2 = 0 \]

2. **Denominator**:
\[ \sqrt{0^2 + (-1)^2 + (-2)^2} = \sqrt{0 + 1 + 4} = \sqrt{5} \]
\[ \sqrt{(-4)^2 + 2^2 + (-1)^2} = \sqrt{16 + 4 + 1} = \sqrt{21} \]

Thus, the denominator becomes:
\[ \sqrt{5} \cdot \sqrt{21} = \sqrt{105} \]

### Step 6: Final calculation for \( \cos \theta \)

Putting it all together:
\[ \cos \theta = \frac{0}{\sqrt{105}} = 0 \]

### Step 7: Determine the angle

Since \( \cos \theta = 0 \), this implies:
\[ \theta = \frac{\pi}{2} \text{ radians} = 90^\circ \]

### Conclusion

The angle \( \angle PRQ \) is \( 90^\circ \).

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