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.
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(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
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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 .
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