Part 1

Question 1

Vectors are called parallel or collinear if

Answer 1

u=av or v=bu

Question 2

Commutativity of vector addition

Answer 2

u+v=v+u

Question 3

associativity of vector addition

Answer 3

(u+v)+w=u+(v+w)

Question 4

additive identity

Answer 4

There is a vector 0 (- Rn such that v+0=v

Question 5

Opposite vector

Answer 5

there is a vector−v ∈ n such that v + (−v) = 0;

Question 6

Distributivity of vector addition

Answer 6

a(u + v) = au + av;

Question 7

Distributivity of scalar addition

Answer 7

(a + b)v = av + bv;

Question 8

Homogeneity of multiplication by scalar

Answer 8

(a·b)v = a(bv);

Question 9

Unitarity of multiplication by scalar

Answer 9

1v=v

Question 10

dot(inner or scalar) product

Answer 10

u•v=x1y1+x2y2

Question 11

Vectors are orthogonal or perpendicular

Answer 11

If the dot product is 0

Question 12

symmetry

Answer 12

u•v=v•u

Question 13

Homogeneity

Answer 13

(au)•v = a(u•v)

Question 14

Distributivity

Answer 14

(u + v)•w = u•w + v •w

Question 15

Positiveness

Answer 15

v •v ≥0, and v•v = 0 if and only if v=
0.

Question 16

norm or vector length

Answer 16

|v|= (v•v)^1/2

Question 17

normalised vector or unit vector

Answer 17

|v|=1

Question 18

Cauchy-Schwarz inequality

Answer 18

|u•v|≤|u|·|v|

Question 19

Triangle inequality

Answer 19

|u + v|≤|u|+ |v|

Question 20

Pythagoras Theorem

Answer 20

|u + v|^2 = |u|^2 + |v|^2

Question 21

The angle between between vectors

Answer 21

cos(ϕ) = u•v/|u||v|.

Question 22

proju (v)

Answer 22

(u•v)u/(u•u)
The projection of v on u

Question 23

vector can be presented using the fixed vectors d and p:

Answer 23

v = p + td

Question 24

parametric form of the line l

Answer 24

x = p1 + td1
y = p2 + td2

Question 25

normal form

Answer 25

n•v = n•p

Question 26

general form

Answer 26

ax + by + c = 0.

Question 27

vector form of the plane P

Answer 27

v = p + td + sk

Question 28

parametric form of the plane P

Answer 28

x = p1 + td1 + sk1 y = p2 + td2 + sk2 z = p3 + td3 + sk3

Question 29

general form (or equation) of the plane P

Answer 29

ax + by + cz + d = 0

Question 30

the real part of x = a + bi

Answer 30

Re(x) = a

Question 31

imaginary part of x = a + bi

Answer 31

Im(x) = b

Question 32

the conjugate of x(complex number)

Answer 32

x ̄ = a − bi

Question 33

modulus (or absolute value) of a complex number

Answer 33

r = |x| = square root a^2+b^2 ∈ [0, ∞)

Question 34

argument of a complex number

Answer 34

θ = arg x ∈ (−π, π]

Question 35

polar or trigonometric form of a complex number:

Answer 35

x = r(cos θ + i sin θ)

Question 36

x = r(cos θ + isin θ) then x ̄ =

Answer 36

r(cos(−θ) + isin(−θ))

Question 37

xx ̄ =

Answer 37

r2 = |x|2

Question 38

De Moivre’s formula

Answer 38

x^n = r^n(cos nθ + isin nθ).

Question 39

Zm contains an inverse x^(−1) ∈ Zm iff

Answer 39

m is a prime number

Question 40

If m = kl is a composite number then

Answer 40

there is no k^(−1) in Zm

Question 41

Field

Answer 41

R, Q, C, Zn