Prestest 2 of Math 1004 Geometry on Monday 03-04/05/2021 Instructions: (i) There are 4 questions in this paper, answer all 4 questions. (ii) Work alone, and write your solution on A4 paper properly. (iii) Submit your solution on Tuesday class. 1. Let Π : ⟨a, x − x0 ⟩ = 0 be a line in R3 , and S = { x ∈ R3 | ∥x∥ = r }. |⟨x0 , a⟩| Prove that Π ∩ S = ∅ if and only if r < . ⟨a, a⟩ 2. Let a ∈ R3 be a non-zero vector. Define a linear map T : R3 → R3 by T (x) = a × x for any x ∈ R3 . (i) Determine the dimension of the kernel ker T of T. (ii) For any b ∈ R3 , prove that b is in the range of T if and only if ⟨a, b⟩ = 0. 3. Let A, B be 2 distinct points in space R3 , and real number λ > 1. # » # » (i) Prove that the set S = { X ∈ R3 | ∥AX∥ = λ∥XB∥ } is a sphere in R3 ; (ii) Determine the center of S and the radius of S. 4. Let P be a point inside the triangle ABC on the same plane such that there exist positive numbers # » # » # » # » α, β, γ such that OP = αOA + β OB + γ OC and α + β + γ = 1. (i) Define X = AP ∩ BC, Y = BP ∩ CA and Z = CP ∩ AB. Determine the ratio of α, β and γ. SXY Z SABC in terms (ii) Define X, Y, Z be points on sides BC, CA, AB such that AP ⊥ BC, BP ⊥ CA and CP ⊥ AB. Z Determine the ratio SSXY in terms of α, β and γ. ABC Hint. P X ⊥ BC ⇐⇒ P B 2 − P C 2 = XB 2 − XC 2 . 1