Let S E R3 I 1113112 = 1} be the unit sphere in . The tangent plane to S at p is the plane through the point p with normal vector p. It is the unique plane which is tangent to the surface S at p. a) Give a scalar equation for the tangent plane at a point p = (a, b, c) in terms of a, b, c. [2]
13(1,1, 1). b) Give a vector equation for the tangent plane at p = 7 Draw a picture illustrating your answer. [2] c) The tangent subspace Wp at p is the plane through 0 with nor-mal vector p. Find an explicit formula in terms of a, b, c, x, y, z for the projection Tp(x) of a vector x E R3 to the tangent sub-space Wp.[2] d) Show that I ITp(x)I 12 = 11×112 (x 10)2 for all x E and deduce that if x E R3 is a given nonzero vector, then the function p i Tp(x) vanishes at exactly two points of S. (Hint. When does equality hold in the Cauchy-Schwarz inequality?) [2] e) BONUS. Find all points p such that Tp(1,0,0) and Tp(0, 1,0) form a basis for Wp. [2]
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