Question

Need E, F, G, and H.

Use the following steps to find the general equation of the plane that intersects the surface f(x, y) ye2x-y+5 at f(-1,3) Choose any vector ū, u # 0, in the xy-plane that is parallel to neither the x-axis nor the y а) axis. Use a cross product to show that u is parallel to neither axis. Find Duf-1,3) b) Choose any vector v, v *0, in the xy-plane that is orthogonal to u. Use the dot product to show that u and v are orthogonal Find Df(-1,3) d) Define U (u, u2, ||ū||Df(-1,3)) where u=u,i +u2j in part (a). e) (v1, v2, ||||D5f(-1,3)) where v = v, i +v2j in part (c) Define V f) g) Find the general equation of the plane that is determined by the vectors Ū & V and that contains the point at f(-1,3) What geometric relationship does the plane have with the surface f(x,y) ye2x-y+5? Һ) C(부.! 부) 7arcllel ho nether arir Va Thes 2x-y+5 2x-7t52ts> e KC DG ft- 37 -3) . ,- + 2 17 Vee hr Terpen dieulos to is V: 2 orthog onat the u and are -2---42 17

Answer 1

Soluhbn we have as u,, u2 | Düfl-1,3) and V vecters and uhich has U eg" of Plane -2-3+S 3 e passing thregt f-,a) (-1,3,3) l'e -42 E4-)i-(-4)+k ( - Gi +2j+k eq of Plane (-, 2,)= C y-3, Z-3 - 6 -6 + 2y-6 +2-3 - o -6x+2y +z -15 = 0 6x-2y+z +15= 0 62-2y-Z +15o Plame 2x-y+S Iurface ol lbe = ye xC3) = 3x2 6 e2x-y+s () = e-3e = -2 -1 Hence Normal uechr at (-1,3,3) le f(-i,3) (, dy, J) (-,3) (6,2,-) Hence Plane obteued tn part (G the Surface I's tangent PLane to at fi,3) Ans