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In Exercises $7-10$ , let $H$ be the hyperplane through the listed points. (a) Find a vector $\mathbf{n}$ that is normal to the hyperplane. (b) Find a linear functional $f$ and a real number $d$ such that $H=[f : d]$$$\left[\begin{array}{r}{1} \\ {-2} \\ {1}\end{array}\right],\left[\begin{array}{r}{4} \\ {-2} \\ {3}\end{array}\right],\left[\begin{array}{r}{7} \\ {-4} \\ {4}\end{array}\right]$$

a) $\vec{n}=\left[\begin{array}{l}{4} \\ {3} \\ {-6}\end{array}\right]$b) $f\left(x_{1}, x_{2}, x_{3}\right)=4 x_{1}+3 x_{2}-6 x_{3}$$d=-8$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 4

Hyperplanes

Vectors

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okay. In this question, you want to find the normal better that is normal to the high percentage covenant by B one, B two, B three and the linear functional, if such that all the author of this whole thing. So first off, who perform actually things again. So v to once we won you go to four minus one gives you three minus two minus minus two gives you 03 mice. One cause you to the three months being one is seven minus one is his sixth minus four minus minus two minus 24 months. One give you re so then your normal vector. You know better. What dick that's about being one win is being so It was 6 to 0 and you know more vector about the three month be one most possible victims there surviving these two out fully. You have three and one plus zero times in to Jesus Syria. So, plus to end to June +33 physical to zero and six n one my spine two and two plus three and 3 to 0. Just looking for anyone you think too negative to end three other three. So that's choose n 3223 and end one easy to shoot. So the reason why would choose minus three is just so I can catch it on the street. There's no other reason why you could do the same thing for in 321 you get the same results in putting it back into this equation. You have six times to give you 12 minus to end. Two plus three times minus treaties. You mice line is it? This is your then upload Modest line gives you three are moving out onto the other side. You have negative too. And Chu is attitude 93 So end too easy with you. Sure. So our mom a vector. It's just negative. 33 over two on two. So if this is a normal vector to get rid of the half, we can save this set All normal vector to be equal to Thio End to 2000 original, which is minus six, three and four. I don't know, I have these ones. This is true and this is minus three. So this is 23 and mine, right? Okay, this is two, 32 and 93. So then this is equal to four, three and minus six. So the reason why we can do this right he had to be scaling is if you think back on two dimensions. If you're no more, Victor was like this. Then if you're playing your planes problem Dick Lecter, this normal vector in this plane, this plane will this line will still be perpendicular to two times Inspector. So that's why you can just simply scale it to be this. So I can't We know that a little victor is full three and my six. Well, then a linear functional, as you expected, is X factor in Vector. So that's four x one plus three x two minus six x three. So then, to find the real number d or you need to do is simply substitute X. Better is XlV one or maybe two? We'll be three. Doesn't really matter. In this case. I'll just prove you won. So be one is equal to are think one Maya's shoe and one transfers So off the one vector What Time's one gives you for three times I minus two gives you minus six minus six mile tied by minus one gives you a negative six. So this is four minus minus four minus six is minus two minus 60 minus. It's so this is the

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