CALCULUS 3 !! Hard problems

Each problem is worth 5 pts of extra credit.

1 Problem

(a) Find an equation of the hyper-plane in R5 that passes through the following five

points

(1, 0, 1, 0, −1)

(1, 2, 1, 1, 1)

(1, 2, 1, −2, −1)

(0, 1, 1, 1, 0)

(1, 0, 3, 1, 0)

(b) Calculate the distance from (1, 1, 1, 1, 1) to the plane defined in (a).

2 Problem

Calculate the volume of the parallelepiped in R4 spanned by

(1, 0, 1, 0,)

(3, 1, 1, 1,)

(9, 2, 1, −2,)

3 Problem

Find the canonical form of the second order curve

4x 2 + xy + 2y2 + x + y = 0 (1)

Write the respective change of coordinates that brings (1) to its canonical form.

Sketch the curve in the original system of coordinates (x,y).

4 Problem

Find the canonical form of the second order surface in R3

2x 2 1 + 2x

2 2 + x

2 3 + x1 · x2 + 2x2 · x3 + x1 + x2 + x3 = 0 (2)

Write the respective change of coordinates that brings (2) into its canonical form.

Sketch the surface in the original system of coordinates (x1,x2,x3).

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