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Numerade Educator

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Problem 69 Hard Difficulty

Use the formula in Exercise 12.4.45 to find the distance from the point to the given line.

$ (4, 1, -2) ; x = 1 + t , y = 3 - 2t , z = 4 - 3t $

Answer

$$x+2 y+z=5$$

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Video Transcript

let's talk about this question video to use a formula to find the distance between a point and a line. So the line is in three D is given in the parametric coordinates. Uh So the formula for the distance is the the magnitude of a Crosby factor over the magnitude of factory where uh a factor is uh is Q R and B. Factor B factor is QP. Now the point is what rescue workers are inward speak. So point P is something which from which we need to find the distance. So that's four comma one comma two minus two. Mhm Q and R. R two points which are lying on this line. So we can easily find out by placing, let's say T as zero and T as one. So if you place tr zero, the value of X, y and Z comes out as 13 and four. And if you place T as one And then the value comes out as 2, 3 minutes to is one and 4 -3 is also one. Uh Now we need to find the factors as well. So the factor Q r is going to be 1 -2 and -3 because we got to subtract the coordinators of our uh we ought to subtract corners of cue from the corresponding corners of our and likewise QB Q p is going to be uh uh three -2 and -6. Now we need to find the cross product, let's rewrite this as you are cross QP over over uh Q R a magnitude. So now we need to find the cross product of Q. R. And Q. P. So that's gonna be I. J. K. We are finding Q. R. Cross QP Than 1 -2 -3. We're just using the formula of the matrix. This is how it looks like. So that's going to be equal to hi at times. And this is gonna be 12 minus six minus J. Times. There's gonna be minus six plus nine. And this is gonna be K. Times -2 Plus six is going to be six I -3 J. Plus four kids. But we are interested in its magnitude. So although this is the absolute wetter cuba across ups, so the magnitude is going to be cuba across U. P magnitude. Which is square root of six square three square four square. Certainly grabbed my calculator here. It's gonna be six square three square and four square. So that's coming as 61. So we have the root of 61 over here. And let's not boxer for the moment. We need to also find the value of the magnitude of Q. Bar. And we already have Q. R. Over here. So let's find that this would be a pretty much straightforward using the same using the formula. So the magnitude of the factor Q. R. Is going to be square root of one square plus two squared plus three square. Which is going to be nine plus four plus +19 plus four is 13 and 13 plus one as 14. So that's gonna be a square root of 14. Uh So finally uh distance D. Is gonna be the magnitude of QP cross Q. R. Which regard as 61 61. I pardon over the magnitude of Cuba, which is the root of 14, so it can be read and a square root of 61/14, which is the required distance at this point from the line. Thank you.