Planes in R3 12 5 Cross Product 12 4 Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Describing a Plane A point P0 on the plane a normal vector n perpendicular to the plane For any other point P on the plane the vector P0 P r r0 is perpendicular to n Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Equations for a plane n a b c P0 x0 y0 z0 Then P x y z is on the plane if and only if 0 a b c x x0 y y0 z z0 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Equations for a plane n a b c P0 x0 y0 z0 Then P x y z is on the plane if and only if 0 a b c x x0 y y0 z z0 That is ax by cz ax0 by0 cz0 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Equations for a plane n a b c P0 x0 y0 z0 Then P x y z is on the plane if and only if 0 a b c x x0 y y0 z z0 That is ax by cz ax0 by0 cz0 Writing d ax0 by0 cz0 we have the equation of the plane ax by cz d Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Example Tangent Plane Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Intersections and Angles General Question Given two planes P1 and P2 how can we see 1 If they intersect 2 If so where do they intersect 3 What is the angle of their intersection Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Example Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Example Planes P1 is the plane given by z 1 P2 is set of points satisfying x z y Normals n1 0 0 1 n2 1 1 1 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Example Planes P1 is the plane given by z 1 P2 is set of points satisfying x z y Normals n1 0 0 1 n2 1 1 1 Intersection is a line L consisting of points satisfying both equations i e x y z so that x z y and z 1 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Example Planes P1 is the plane given by z 1 P2 is set of points satisfying x z y Normals n1 0 0 1 n2 1 1 1 Intersection is a line L consisting of points satisfying both equations i e x y z so that x z y and z 1 For example 1 0 1 0 1 1 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Angle Angle between planes is the angle between their normal vectors In our example we have cos 0 0 1 1 1 1 1 n1 n2 kn1 kkn2 k k 0 0 1 kk 1 1 1 k 3 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Vectors in R3 Cross product is a way to combine vectors u v in R3 to get a new vector u v which is perpendicular to both of them It s very useful for describing planes Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinant a b Given a 2 2 matrix we define the determinant by c d a b a b det ad bc c d c d Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinant and Area Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinant Angle and Area w h w c d v a v b Area A base height kvkkwk sin Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinant Angle and Area A2 kvk2 kwk2 sin2 kvk2 kwk2 1 cos2 kvk2 kwk2 v w 2 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinant Angle and Area A2 kvk2 kwk2 sin2 kvk2 kwk2 1 cos2 kvk2 kwk2 v w 2 v a b w c d so we get A2 a2 b2 c 2 d 2 ac bd 2 a2 d 2 b2 c 2 2abcd ad bc 2 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinant Angle and Area A2 kvk2 kwk2 sin2 kvk2 kwk2 1 cos2 kvk2 kwk2 v w 2 v a b w c d so we get A2 a2 b2 c 2 d 2 ac bd 2 a2 d 2 b2 c 2 2abcd ad bc 2 a b ad bc c d 2 2 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinants and Rows Also for any t R a b a b a tc b td c ta d tb c d c d Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinants and Rows Also for any t R a b a b a tc b td c ta d tb c d c d Exercise think of what this means in terms of parallelograms reduce area computation to the case of a rectangle Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Signed Area 1 2 3 1 1 3 3 1 2 1 6 5 Why negative 1 1 6 1 5 2 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Signed Area 1 2 3 1 1 2 1 6 5 Why negative 3 1 3 1 6 1 5 1 2 Reason sin sin Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Signed Area a b c d AREA AREA if a b to c d is counterclockwise if a b to c d is clockwise Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinant in 3 dimensions u1 u2 u3 v v2 v v3 v v3 v1 v2 v3 u1 2 u3 1 u2 1 w1 w2 w1 w3 w2 w3 w1 w2 w3 Jayadev S Athreya Math 241 Spring 2014 Planes in R3 12 5 Cross Product 12 4 Determinant in 3 dimensions u1 u2 u3 …
View Full Document
Unlocking...