Use vectors to decide whether the triangle with vertices
P( 1, -3, -2) , Q (2, 0, -4), and R( 6, -2, -5) is right-angled.

To determine if the triangle with vertices \( P(1, -3, -2) \), \( Q(2, 0, -4) \), and \( R(6, -2, -5) \) is right-angled, we can use the concept of vectors and the dot product.

### Steps:
1. **Find the vectors representing the sides of the triangle:**
   - Vector \( \overrightarrow{PQ} \) from \( P \) to \( Q \):
     \[
     \overrightarrow{PQ} = Q - P = (2 - 1, 0 + 3, -4 + 2) = (1, 3, -2)
     \]
   - Vector \( \overrightarrow{PR} \) from \( P \) to \( R \):
     \[
     \overrightarrow{PR} = R - P = (6 - 1, -2 + 3, -5 + 2) = (5, 1, -3)
     \]
   - Vector \( \overrightarrow{QR} \) from \( Q \) to \( R \):
     \[
     \overrightarrow{QR} = R - Q = (6 - 2, -2 - 0, -5 + 4) = (4, -2, -1)
     \]

2. **Check if any pair of these vectors are perpendicular by calculating their dot products:**
   - The dot product of \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \):
     \[
     \overrightarrow{PQ} \cdot \overrightarrow{PR} = (1)(5) + (3)(1) + (-2)(-3) = 5 + 3 + 6 = 14
     \]
   - The dot product of \( \overrightarrow{PQ} \) and \( \overrightarrow{QR} \):
     \[
     \overrightarrow{PQ} \cdot \overrightarrow{QR} = (1)(4) + (3)(-2) + (-2)(-1) = 4 - 6 + 2 = 0
     \]
   - The dot product of \( \overrightarrow{PR} \) and \( \overrightarrow{QR} \):
     \[
     \overrightarrow{PR} \cdot \overrightarrow{QR} = (5)(4) + (1)(-2) + (-3)(-1) = 20 - 2 + 3 = 21
     \]

### Conclusion:
The dot product \( \overrightarrow{PQ} \cdot \overrightarrow{QR} = 0 \), which indicates that vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{QR} \) are perpendicular. Since two sides of the triangle are perpendicular, the triangle is right-angled.

Would you like further details or have any questions about this solution? 

Here are some related questions to explore further:
1. How can the magnitude of the vectors be calculated, and how is it relevant to the problem?
2. What if the triangle was in a different plane? Would the method change?
3. Can the same process be applied to check for obtuse angles in the triangle?
4. How would the method differ if the vertices were given in 2D?
5. Can you find the area of this triangle using vector methods?

**Tip:** The dot product of two vectors is a quick way to determine if they are perpendicular. If the result is zero, the vectors are orthogonal.

Explore how vectors and dot products are used to decide if a triangle with vertices P(1, -3, -2), Q(2, 0, -4), and R(6, -2, -5) is right-angled. Learn the step-by-step method, including calculating dot products and interpreting results. Suitable for high school students studying vector algebra.

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