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Problem 72 Hard Difficulty

Find the distance from the point to the given plane.

$ (-6, 3, 5) , x - 2y - 4z = 8 $

Answer

Distance $=\frac{40}{\sqrt{21}}$

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Video Transcript

the question they're asking to find the distance from the point to the plane. The point is minus 635. And the plane equation is given by x minus two, Y minus four. Difficult to eight. Yeah. Yeah. All the time. Okay. Yeah. Therefore in order to find the distance from the point to the plane you need to find the normal vector to the plane. So here normal vector to the plane is equal to no Coefficient of the experience. That coordinates suspected besides equal to 1 -2 and -4 which is equal to a victor I suppose and a point. So with the plane passes. Uh huh. Okay. Okay. Yeah. Yeah. Can be found out by putting any values of the variables, like for example X equal to why? What does the report these two reporters zeros then the values that can be found out from the creation. So if we put x y zero nick question we get minus for the record eight. Therefore in this situation we get the values that you go to -2. Therefore a point we got a point Into with the pain passes. That is equal to 00 -2. At the point be denoted by the variables Q. Now, since uh yeah, we have to find of vector B victor. Which yeah. Which is the factor found by calculating the difference between Q and a pre victor. If you find this value then we get value of the factor which is lying Mhm. On the line, in which the point connects with the plane. So this is equal to let's look- is that if you go to 6 -3 -7 therefore the formula to calculate the distance from the point B. Mhm. Let's let's suppose the Point B. With this -635 has given in the question then all then to the plane asked during the question the question of the plane. So the formula body is here is equal to a vector dot be vector divided by a vector whole scalar value. So the value of a vector or the victories equal to 1 -2 -4. Thank you. No this equal do the value of me Back to 6th -3 -7. Yeah. That possibility this divided by route under. So in the left hand side read the scalar value of director. That is here. The a victor is 12 and 41 minus two and minus photo. So it will be wanting to see purpose. Foursquare So this entire time will be equal to the dot product is the multiplication of each other coordinates Of the two vectors and their son. So this is a quick six last 6. 28 divided by mhm. Mhm. Route and 21. This is equal to 40. The very valued under 21 hens. The answer to the Cuban question is the distance from the point B. Mhm. Okay. That is given in the question to be yeah. What's up -635. And the question of the plane. That is given by the equation next minister, Y -4 equals eight. So the distance between these two entities that is a point and a plane is given by the formula A vector not be vector by a vector scalar value. So Therefore after evaluating the value of D, So D is equal to 40. Mhm. Divided by Route UNDER 21. Mhm. Therefore this is the required answer. I'll take you any question.