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sampletext5.txt
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440 12.5 EQUATIONS OF LINES & PLANES > normal vector determines a uniquep equations : Planes "a plane in space Is determined by a point In the plane and a normal which is Il to and two vectors on the plane . note that - a, b, c - components of v eq xy plane ? use origin n. r = n. ro - r ( r -ro ) = 0 < a, b, ( > . < X - xo , Y - YO , 2 -20 > =0 a ( x - xo ) + b ( 4 - Yo ) + ( ( 7-20) = 0 Vector Equation of a Plane Scalar Equation of the Plane linear equation of a plane determine the general form of the eq. a x + by + cz + d = O / = d > use n _ n, x + n 2 y + n 3 z = d > Plug in valves of Po for variables General Form for the Equation of a Plane > solve for d, n, x + n, y rn, 2 = d IF the three coll , a planes contain them Use this form to make systems to find & to find the normal u when only given three points, you can choose po, , and Pz 1) determine POP, [ PO - P. ] 2) P. P, x P. P, which gives you P . - POP POR x POP. the normal vector. the equation follows : n, (x - P ., ) ... > planes are Il when n, IIn. , Perp when n, n2 = 0 POPE P2 PO Note : use parametricea to show / check that the line lies in eachplane. (Intersection) you can also show line A plane, soIntersects Ipt only determine distance from given pr to given piane : choose an arbitrary point P. In given plane then find the scalar projection OF POP. Onto A. Icomp, Pop, I = In. Pop. . dist blw planes make like y = = = o and solve for X, plug into equation shown below. . planes must x when they aren't Il . determine plane when given point and line: we need at least two v, so use the pr- . Substitute a random value for + to get another et, then find v between those pts and Xmultiply . to find V with symmetric eg of line, Set = to zero. . When given two planes and need to find line OF Intersection : cross the two normal vectors OF P's . Xint > n, = d etc ... Il find the pt @ which the line intercepts plan, sub all x, y? values Into eq of plane, solve for + and plug back Into eq of a line find eq for plane consisting of all DIS equidistant from given Points # 61 ( 872)