Eureka Math Algebra I Study Guide

116 | eureka Math algebra I Study guIde

● (^) Opposite Vector For a vector



v represented by the directed line segment AB



, the

opposite vector, denoted -



v, is the vector represented by the directed line segment BA



.

If





v

v

v

vn

=

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú

1

(^2) in ℝn, then -=

-

-

-

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú





v

v

v

vn

1

(^2).
● (^) Polar Form of a Complex Number The polar form of a complex number z is
ri(cos()qq+ sin( )) where rz= and q=arg(z).
● (^) Position Vector For a point P(v 1 , v 2 ,.. ., vn) in ℝn, the position vector



v, denoted by

v

v

vn

1

2



é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú

or ⟨v 1 , v 2 ,.. ., vn⟩, is a free vector



v that is represented by the directed line segment OP



from the origin O(0, 0, 0,.. ., 0) to the point P. The real number vi is called the ith

component of the vector



v.

● (^) real Coordinate Space For a positive integer n, the n-dimensional real coordinate space,
denoted ℝn, is the set of all n-tuple of real numbers equipped with a distance function d
that satisfies
dx 12 xxnnyy 12 yy 11 xyxynnx
2
22
2
éë(),,¼,,(),,¼, ùû=-()+-()++ ()-
22
for any two points in the space. One-dimensional real coordinate space is called a
number line, and the two-dimensional real coordinate space is called the Cartesian plane.
● (^) rectangular Form of a Complex Number The rectangular form of a complex number z is
ab+ i where z corresponds to the point (a, b) in the complex plane and i is the imaginary
unit. The number a is called the real part of ab+ i, and the number b is called the
imaginary part of ab+ i.
● (^) Translation by a Vector in real Coordinate Space A translation by a vector v in ℝn is the
translation transformation Tv:ℝℝnn® given by the map that takes



xx +v for all



x in

ℝn. If





v

v

v

vn

=

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú

1

(^2) in ℝn, then T
x
x
x
xv
xv
xv
v
nnn


1
2
11
22
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
æ
è
ç
ç
ç
çç
ö
ø

÷

÷

÷

÷÷

=

+

+

+

é

ë

ê

ê

ê

êê

ù

û

ú

ú

ú

ú

for all



x in ℝn.

● (^) Vector Addition For vectors v and w in ℝn, the sum vw+ is the vector whose ith
component is the sum of the ith components of



v and



w for 1 ££in. If





v

v

v

vn

=

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú

1

(^2) and





w

w

w

wn

=

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú

1

(^2) in ℝn, then 

vw
vw
vw
vwnn

+=

+

+

+

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú

11

(^22).