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Find the values of $ x $ such that the angle between the vectors $ \langle 2, 1, -1 \rangle $, and $ \langle 1, x, 0 \rangle $ is $ 45^\circ $.

$x=1 \pm \frac{\sqrt{6}}{2}$

Vectors

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Okay, so let's call this vector here. A It's called Inspector. Here, be this guy angle. See? So okay dot B is equal to a magnitude. Be magnitude co sign Ada. Okay, so we need to plug this stuff in. So let's do a dot b is going to be two plus x. Okay. Magnitude of a he's going to be squared of two squared plus one squared plus negative one squared. So this is going to be rod six, and that's a magnitude of B is going to be, uh, end of being one plus x squared. Okay, swear to that. Okay, so planning all this into this formula we get to plus X is equal to rad six. Was this then co sign of forty five to read too? Okay, So what we're gonna do is we're going to square both sides of this equation. Hey, squaring both sides, get four plus four x plus. Vex swear plus IQ is equal to six plus six x. And then if we swear this, we get one half. Okay, so then, Mr Street right that four plus for X plus X squared is equal to one half of six plus six X squared is a quadratic equation, so that four plus four x plus x squared is equal to three plus three. Ex swear now going to move some stuff around. So I'm going to move everything to the side and we're going to get to X squared minus four X minus. One is equal to zero. Yeah, just double check that. Okay. Great. And then we're going to use a quadratic formula here. So it's equal to four plus or minus sixteen plus for a C two four times two eight all over two times two with four plus or minus squared of twenty four over floor for plus cheer at six over four. And then so we can simplify this as two plus are well not to. Uh, yes. And we don't divide by two plus or minus. Tried six all over too. So that's our values for hands