Problem 82 Hard Difficulty

Give a geometric description of each family of planes.

(a) $ x + y + z = c $ (b) $ x + y + cz = 1 $
(c) $ y \cos \theta + z \sin \theta = 1 $

Answer

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WZ

Wen Z.

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Video Transcript

the question uh they are asking to give a geometric description of each other family of planes. Question it the question is expressed by place article two C. B. Expressed by Percy is equal to one. And see why he caused me to present scientific equal to one. So for the question A. The planes X plus Y plus it equal to see have normal victor 111. So they are all parallel. Their X. Y. And Z intercepts are all see When C. is greater than zero. Their intersection with the first top tent in an equal lateral triangle. Their intersection with the first opt in and is an equilateral triangle. Yeah. And when See is less than zero. That intersection with the extent diagonally opposite the first is an equal lateral strangle. Mhm. That can be presented in the oh picture in the diagram with the coordinates Zed X and why? So the intercept R. C. C. And seeing all the access therefore the equilateral triangle formed with these intercept is this and Mhm. Yeah this is the opt in so or this answer to the question A And next in the second question they're asking that uh for our equation um to interpret its geometric description. So the planes express why Plus season is equal to one. Have have a X intercept one. Why intercept 1? And they intercept see That intercept one by sea. Therefore the plane with c equal to zero is parallel to the the access as he gets larger. Then the planes get closer to them. Xy plane. This is the answer of question be And in uh the equations see the planes why cost a to plus said sign people Is equal to one. So this plane have normal vectors. Zero. Yeah. Cause theater and scientific. No. Now this is perpendicular to X axis. And so the planes are parallel to the X axis. We look at the intersection with the wiser plane. Mhm. These are lines that are perpendicular to the vector cost data. Scientific. Oh and path through because he doesn? T to since or square tita less sine squared theta Is equal to one. So these are the tangent lines to the unit circle. Thus the family consists of all planes tangent to the circular cylinder. Oh with radius one and access. Yeah. Is the X axis. So this is the required answer of the given question.