8/7/2019 cwgeom111

1/5

Introduction to Geometry (20222)

2011

COURSEWORK

This assignment counts for 20% of your marks.

Solutions are due by 1-st April

Write solutions in the provided spaces.

STUDENTSS NAME:

1

8/7/2019 cwgeom111

2/5

1

a) Let (x1, x2, x3) be coordinates of the vector x, and (y1, y2, y3) be coordinates of

the vector y in R3.

Does the formula (x, y) = 2x1

y1

+ x2

y2

+ x1

y3

+ x3

y1

+ 2x3

y3

define a scalar product

on R3? Justify your answer.

b) Vectors a, b, c, d in Euclidean space are pairwise orthogonal to each other and all

of them have non-zero length.

Prove that these vectors are linearly independent.

What can you say about a dimension of this Euclidean space?

c) Let a, b be two vectors in the Euclidean space E2 such that the length of the vector

a equals to 3, the length of the vector b equals to 5 and scalar product of these vectors

equals to 9.

Show that these vectors span E2.

Consider the pair of vectors {x, y} such that {x, y} = {a, b}T, where transition matrix T

equals to T = T1T2 and

T1 = 1

3

1

4

0 1

4 , T2 = cos sin

sin cos , is an arbitrary angle .

Find lengths of the vectors x and y and their scalar product.

2

8/7/2019 cwgeom111

3/5

2

a) Consider the matrix

A =

cos sin sin cos

.

Calculate the matrix A2 in the case if = 4

.

Calculate the matrix A18 in the case if = 6

.

Calculate the matrix A2011 A in the case if = 267

.

b) Let {e1, e2, e3} be an orthonormal basis in 3-dimensional Euclidean space E3. Let

P be a linear operator acting on 3-dimensional Euclidean space E3, such that Pe1 = e2,

Pe2 = e3 and Pe3 = e1.

Show that for arbitrary two vectors x, y E3 (x, y) = (Px, Py) where (x, y) is a

scalar product in E3.

Find a non-zero vector f such that Pf = f.

What is a geometrical meaning of this vector?

c) On the plane OX Y find the horizontal line l: y = c and the point F = (0, f) on

the OY axis such that for all the points M on the parabola y = x21

2the distance |M F|

equals to the distance between the point M and the line l.

3

8/7/2019 cwgeom111

4/5

3

a) Consider vector a = 2ex + 3ey + 6ez in E3, where {ex, ey, ez} is an orthonormal

basis for Euclidean space E3

Show that the angle between vectors a and ez belongs to the interval6 , 4

.

Find a unit vector b such that it is orthogonal to vectors a and ez, and the bases

{ex, ey, ez} and {b, ez, a} have opposite orientation.

b) In oriented Euclidean space E3 consider the following function of three vectors:

F(X, Y, Z) = (X, Y Z) ,

where ( , ) is the scalar product and Y Z is the vector product in E3.

Show that F(X, X, Z) = 0 for arbitrary vectors X and Z.

Deduce that F(X, Y, Z) = F(Y, X, Z) for arbitrary vectors X, Y, Z.

What is the geometrical meaning of the function F?

c) Consider a triangle ABC in E3, formed by the vectors a = (85, 48,36) and

b = (84, 48,36) attached at the point C.

Calculate the area of the triangle ABC.

Calculate the length of the height CM of this triangle. (CM AB and the point M

belongs to the line AB.)

Show that the vector h = CM and the vector c = (a b) (a b) are collinear

(proportional).

4

8/7/2019 cwgeom111

5/5

4

a) Given a vector field G = ar r

+ b

in polar coordinates express it in Cartesian

coordinates (x = r cos , y = r sin ).

b) Consider the function f = r2

sin2 and the vector fields A = xx + yy, B =

xy yx.

Calculate Af, Bf.

Calculate the value of 1-form df on the vector field 3A + 2B.

Express all answers in polar and in Cartesian coordinates.

(c) Show that for 1-form = 3(x2 y2)dx 6xydy there exists a function f such

that = df (i.e. is an exact 1-form).

Express the 1-form in polar coordinates.

Calculate the value of the 1-form on the vector field A = r cos3 r sin3

.

5