To determine whether the triangle with vertices $P(1, -3, -2)$, $Q(2, 0, -4)$, and $R(6, -2, -5)$ is right-angled, we can use vector properties. Specifically, we will check if any pair of the vectors representing the sides of the triangle is perpendicular by verifying if their dot product is zero.

Step 1: Find the vectors representing the sides of the triangle.

The vectors corresponding to the sides of the triangle are:

• $\overrightarrow{PQ} = Q - P = (2 - 1, 0 - (-3), -4 - (-2)) = (1, 3, -2)$
• $\overrightarrow{PR} = R - P = (6 - 1, -2 - (-3), -5 - (-2)) = (5, 1, -3)$
• $\overrightarrow{QR} = R - Q = (6 - 2, -2 - 0, -5 - (-4)) = (4, -2, -1)$

Step 2: Calculate the dot products of these vectors.

For two vectors to be perpendicular, their dot product must be zero. Let’s check the dot products between the pairs of vectors.

1. Dot product $\overrightarrow{PQ} \cdot \overrightarrow{PR}$:

$\overrightarrow{PQ} \cdot \overrightarrow{PR} = (1 \times 5) + (3 \times 1) + (-2 \times -3) = 5 + 3 + 6 = 14$ Since $14 \neq 0$, $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ are not perpendicular.

2. Dot product $\overrightarrow{PQ} \cdot \overrightarrow{QR}$:

$\overrightarrow{PQ} \cdot \overrightarrow{QR} = (1 \times 4) + (3 \times -2) + (-2 \times -1) = 4 - 6 + 2 = 0$ Since the dot product is $0$, $\overrightarrow{PQ}$ and $\overrightarrow{QR}$ are perpendicular.

3. Dot product $\overrightarrow{PR} \cdot \overrightarrow{QR}$:

$\overrightarrow{PR} \cdot \overrightarrow{QR} = (5 \times 4) + (1 \times -2) + (-3 \times -1) = 20 - 2 + 3 = 21$ Since $21 \neq 0$, $\overrightarrow{PR}$ and $\overrightarrow{QR}$ are not perpendicular.

Conclusion:

Since $\overrightarrow{PQ}$ and $\overrightarrow{QR}$ are perpendicular, the triangle $PQR$ is right-angled at vertex $Q$.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. How would you find the area of the triangle given the vertices $P$, $Q$, and $R$?
2. How can you determine whether a triangle is isosceles using vectors?
3. What is the geometric interpretation of the dot product in relation to vectors?
4. How do you find the angle between two vectors using the dot product?
5. How can the cross product be used to determine the area of a triangle?

Tip: The dot product of two vectors provides insight into the angle between them: if the dot product is zero, the vectors are perpendicular.