Homework 5, Analytic Geometry and Matrices Determination of Traces

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Homework 5, Analytic Geometry and Matrices

Determination of Traces:

1. Each defining equation below corresponds to only one possible trace from sketches a through l below. Match it and provide sufficient reasons.

(a) x = 2z²− 2y².

(b) 9x²+ 4y²+ 2z² = 36.

(c) x = −y²− z². (d) z = −2x²− y².

More Change of Coordinates:

1. Find appropriate change of coordinate that turns the regions below into regions of con- stant bounds (i.e. rectangles or boxes)

(a) 9x²+ 16y²+ 4z²− 8z − 140 ≤ 0 and z ≥ 1.

(b) Region in R² bounded by y₁ = 3x+8, y₂ = −2x+7, y₃ = 3x+12, and y₄ = −2x+10.

2. A 1-torus in R³ is formed by rotating the circle on yz plane with radius 1 and center (0, 2, 0)

around z-axis to form a ”donut”:

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(a) Construct a change of coordinate that turns the torus into a square [0, 2π) × [0, 2π).

(Hint: Rotate the circle on yz plane by θ, around z-axis and figure out another angle φ so that (θ, φ) determines the points on the torus.)

(b) Sketch the images of θ = π and φ = ^π₂ on the torus under this change of coordinate.

(c) Construct a change of coordinate that turns the region bounded by torus into a box [0, 2π) × [0, 2π) × [0, 1].

Vectors in R³:

1. Prove that two lines having two distinct points in common must be the same line. That is, x 6= x⁰ and x, x⁰ ∈ L ∩ L⁰ ⇒ L = L⁰. (Use the definition of lines and conditions of equality stated in class.) Use this fact to prove that the line (1, −1, 0) + R(1, 1, 1) is the same as the line given by the intersection of two planes x + y − 2z = 0 and 2x − y − z = 3.

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