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Consider the vector-valued function #(t) = (VE,t-3,t2} 1 , Find set of parametric equations for the line tangent to 7 atthe point A(2,1,16) We"Il call this lin...

Question

Consider the vector-valued function #(t) = (VE,t-3,t2} 1 , Find set of parametric equations for the line tangent to 7 atthe point A(2,1,16) We"Il call this line L: 2. Compute the angle between the line L and the plane P:x - 22 = 4 You may leave your answer in terms of an inverse trig function.

Consider the vector-valued function #(t) = (VE,t-3,t2} 1 , Find set of parametric equations for the line tangent to 7 atthe point A(2,1,16) We"Il call this line L: 2. Compute the angle between the line L and the plane P:x - 22 = 4 You may leave your answer in terms of an inverse trig function.



Answers

Find an equation for the tangent plane and parametric equations for the normal line to the surface at the point $P .$ $$ x^{2} y-4 z^{2}=-7 ; P(-3,1,-2) $$

Parametric equations of the tangent line to this curve X. Y. Z. At the 0.21. So for sonny dX DT which will be one over T. An dy DT, which will be two times one half T. To the minus one half Which is one over the square root of T. And then DZ DT equals two teeth. Okay? So at the 0 to 1 um we have t squared equals one. So T. Is plus or minus juan. But why is two squirts of tea? So T. Must be positive one. So these slopes become one one and what's two? So x equals zero plus 1. T. Y. equals two plus 1. T. Z equals I can't see what that oh two plus one cheek. Wait X's wine. Okay. See is one uh just got all confused there. One because that the uh in the point the z coordinate is one and then plus two T. Because that's the slope. So one T. Two plus one T. One plus two T.

Pacifying Parametric equations for line tangent to the curve of intersection of the searches at the given point. So first question, we have two different surfaces and just make our lives easier. One. Go ahead, label them. So I'm gonna have the 1st 1 bf and that is X plus y squared Plus Z is equal to two. I'm a 2nd 1 BG, and that is why it was one we're giving the point 1/2 won. It's not half. So we want to find Parametric questions for lying tangent to the curb of the intersection of these two surfaces. So I already know that the tangent line is a funnel to both the grating of up and the great and G our initial point. And so then therefore it's going to be parallel to the vector. B is equal to grave across Grady and G. And so the component of E and the coordinates of peanut will give us the equations of the line. So what? I'm going to need a fine first. Is it? So first off. Nor have this I didn't know my grade of F and great Angie is so the grating of F evaluated at my point sequel to celebrating it. That's just the partial derivatives added together, So partial derivative with respect to X first is just gonna be one someone I one times my vector I when I take partial derivative with respect to why I just get to why don't take it with respect to Z again. Just get one valuing this at my initial point. So the only variable that comes up here is this Why so why is just equal to one here? So I get just I plus to J less came and then doing the same thing. So for my second equation, well, here I just have y equals once the only partial derivative I can take of this. That's a knock and equal zero. It's just the partial derivative with structural. Why not just give you one again, Ryan taking with respect, Why did J Vector is what goes along with that. And so when I evaluate this up the point since there's no variables in there, it's just gonna be J. So now I have my Grady INTs and I confined my vector B so that's just gonna be I plus TJ. Let's k cross J which is equal to the terminus of I Okay, coefficients of my first. So once you won in the coefficients of my second 01 Yeah. And when I say those determinant, I get negative. I plus Okay, so now that I have my V, I can get my equation. I know that my equation is parallel to this, so I just use what we've done before all these. They're gonna start with my ex, not Why not? See, Not my ex knots. 1/2 my Why not? Is one Messina is 1/2 and then to get the coefficient of tea. I'm just gonna take the coefficients of this since I know it's parallel to this vector. So the coefficient from my accident minus one so small my s t. There's no Jay term here, So there's no t turn for my Why In coach coefficient of K, it's just one B plus t. Yes, that's my answer.

Morning problem. We're going to find the equation for the tension spring on the Parametric equations for the normal line with different functions and to give you fun. Now we can accomplish those. Are you drinking? E have to the right of the screen. I'm not here. What you you need to be. I expect that explains, in order to get the equation for the attendant point on the Parametric equations. So to distill First, let me write the equation. Overhand one, huh? I'm seven, right? Negative minus B Therefore, when it when you find a fashion treat Hawaiian Bs So what we have over to one year is a lonely, hungry, uh, one white floor in what we have, It's something that far can be really respect. Why way have, like again? Well, why you while reading where is too quiet for gets us negative too. And finally f i m c would be equal to negative one kit. So now we can used our formula to get attention mind so we have expects which is 14 times the quantity X minus two. Okay, plus if somewhere just negative two times a quantity wine minus or it's obscene. That's negative born times oak went the quantity the miners for not subzero appear and this is all equal to zero. They distribute. We have 14 X minus 28 minus two. Y ain't minus C plus four equals zero. Combining like terms, we have an equation for detention line or 14 minus two. Why minus thing is equal to 16. Prosthetic 16. So this is the equation for the tension line for the parametric equations who just substitute the values here. So for the first time, we have X equal except zero to f sub X s 14 times t. Why would be equal to why subzero for minus to T and Z would be equal to zero, just four minus? And is that a Parametric equations four to no.


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