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.
Okay, so we want to figure out if this is a right angled triangles. So we need to take the dot product between each one of these. So the dot product between P and Q is going to be two times one. That's negative. Three times zero. It's negative. Two times negative. Four. Okay, After we do this, we get negative. Six. So remember, if the dot product is zero, then we have a right angle, so we don't have a right angle yet. Let's do P and R. So we've got one time. Six plus negative. Three times negative. Two plus negative. Two times negative. Five. Okay, so here we're going to end up with six plus six, which is 12 plus 10, which is 22. So again, still no. Um, right angle. So let's take the dot product between p Sorry, Q and R. So we got two times six plus zero times negative two and then plus negative four times negative five. Yeah, And again, we end up with 12. Plus, uh, I'm sorry. 12 plus 20. Just 2022. So no right angle. So this is no