12.5 Lines and Planes Review: v aaa222 va1,, a 2 a 3 a 1 i a 2 j a 3 k 1 2 3 u v u1 v 1 u 2 v 2 u 3 v 3 u v cos u u u u and v are orthogonal if and only if u v 0 uv uv compvu projvuv v vv u v uv23 uv 32 i uv 13 uv 31 j uv 12 uv 21 k u v is orthogonal to both u and v. u v u v sin u and v are paralallel if and only if u v 0 u varea of the parallelogram determined by u and v Volume of parallelepiped a b c Vector equation of a line L: r r0 t v rv0 point on the line, direction abc, , Parametric (scalar) equation of a line L: xx0 aty, y 0 btz, z 0 ct Planes In order to find the equation of a plane, we need : this vector is called A a point on the plane P0 x 0, y 0 , z 0 the normal vector B a vector that is orthogonal to the plane n abc, , to the plane n r r0 0 vector equation of the plane n r r0 0 a x x0 b y y 0 c z z 0 0 scalar (or component) equation of the plane r0OP 0 x 0,, y 0 z 0 r OP x,, y z n abc,, ax by cz ax0 by 0 cz 0 P0 P r -,,r 0 x x 0 y y 0 z z 0 Plane: ax by cz d Problem : Determine the equation of the plane that passes through PQR(1,2,3), (3,2,1) and ( 1, 2,2) For a point in the plane we can choose e.g. P = (1,2,3) For a vector normal to the plane we can choose the cross product of two vectors in the plane: For two vectors in the plane we can choose e.g. u PQ 2,0, 2 and v=PR 2, 4, 1 i j k uv 2 0 2 08 i 42 j 80 k 8i 6 j 8 k 2 4 1 n4 i 3 j 4 k is normal to the plane 4(x 1) 3( y 2) 4( z 3) 0 4x 3 y 4 z 10 Two distinct planes in 3-space either are parallel or intersect in a line. z y x Question: How do I decide if two planes are parallel? Answer: The normal must be parallel. Are the planes 2x y z 1 and 4x 2 y 2 z 3 parallel? Problem : Find the line of intersection of the two planes x2 y z 0 and 2 x 3 y 2 z 0 2x 4 y 2 z 0 2x 3 y 2 z 0 40xy yx 4 Take the first equation: x2 y z 0 and plug in y 4 x x24x z 07 z x x can be anything and we have y and z as functions of x. Let xt then y4 t and z7 t L: x t, y 4 t , z 7 t If two planes intersect, then you can determine the angle between them. between planes between their normal vectors n1 n2 nn cos 12 nn12 Example: Find the angle between the planes x20 y z 6 54 2x 3 y 2 z 0 102 nn12 1, 2,1 and 2,3, 2 nn126 and 17 nn12 2 6 2 6 Distance between a point and a plane: Q D The point is given by Q b The plane is given by a point P0 and a normal n n Project the vector b PQ0 onto the normal direction n : ||bn P Distance D | compb | 0 n n Example: ||bn Find the distance from Q(1,2,0) to the plane 2 x y 1 D P (0,1,0) n 0 n 2,1,0 b = PQ0 1,1,0 | 1,1,0 . 2,1,0 | 3 D 4 1 5 Distance between a point and a line: The point is given by S The line is given by a point P and a direction v ||PS v Distance D | PS | sin( ) | v | Example: ||PS v Find the distance from S (1,0,1) to the line containing D PP01(1,1,1) and (2, 1,1) | v | i j k 1 PS0 0, 1,0 v PP01 1, 2,0 PS0 v 0 1 0 k D 1 2 0 5 12.6 Quadric surfaces Review conic sections in the plane: x intercept: x a xy22 Equation of an ellipse: 1 y intercept: y b ab22 circle: ab 2 Equation of a parabola: y ax xy22 x intercept: x a Equation of a hyperbola: 1 ab22 no y intercept! x2 y2 z 2 Ellipsoid: 1 sphere: abc a2 b2 c2 Slice the surface with planes parallel to the yz- plane: y2 z 2 k 2 Set x constant: x k, then 1 b2 c 2 a 2 All traces are ellipses a k a Hyperboloid of one sheet: x2 y 2 k 2 x2 y 2 z 2 zk : 1 any k! 1 a2 b 2 c 2 abc2 2 2 Horizontal traces are ellipses x2 z 2 k 2 yk : 1 a2 c 2 b 2 Vertical traces are hyperbolas Hyperboloid of two sheets: zk x2 y 2 z 2 1 2 2 2 222 x y k a b c 1 only kc ! a2 b 2 c 2 Horizontal traces are ellipses Vertical traces are hyperbolas (Elliptical) Paraboloid: z x22 y Vertical traces are parabolas c a22 b Horizontal traces are ellipses (Hyperbolic) Paraboloid: Horizontal traces are hyperbolas z x22 y Vertical traces are parabolas c a22 b Cone z2 x 2 y 2 c2 a 2 b 2 Horizontalx k traces hyperbolas are ellipses k 0 y k hyperbolas k0 lines z y z x , c b c a Vertical traces are hyperbolas Cylinders: xy22 yx22 y ax2 1 1 ab22 ab22 VII IV II III VI I VIII V Draw a picture of the following surfaces: x y22 z a) x2 y22 3 z or elliptic paraboloid 6 3 2 opens in the positive x direction yz22 b) 4 x y22 4 z 0 or x 41 hyperbolic paraboloid Draw a picture of the surface : 4x2 y 2 4 z 2 4 y 24 z 36 0 4x2 y 2 4 y ___ 4 z 2 6 z ___9 36 ___4 ___36 4x2 y 222 4 z 3 4 22 x2 yz23 1 1 4 1 ellipsoid center: 0,2,3 notice e.g. y 0 Summary: 2 2 2 x y Ellipsoidz 22 2 2 2 2 1 z2x x 222 y y 2 za2 b2 c2 xz y x z y Hyperboloid1 of one sheet 2222 2 2 1 all variables present abcc2ca a a 222 b b b 2 c Hyperboloid of two sheets all variables squared Cone elliptic Paraboloid elliptic z x22 y one variable Paraboloid hyperbolic 22 c a b not squared one variable not present cylinder opening in the direction of the missing variable 22 22 xy xz 2 1 Elliptic cylinder 1 Hyperbolic cylinder z ax Parabolic cylinder ab22 ab22.