Eureka Math Algebra I Study Guide

terMInology | 121

● (^) 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=-()+-()+¼+-()
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 A vector is described as either a bound or free vector depending on the context.
We refer to both bound and free vectors as vectors throughout this module.
● (^) 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).
● (^) Vector Magnitude The magnitude or length of a vector



v, denoted



v or



v, is the length of

any directed line segment that represents the vector. If





v

v

v

vn

=

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú

1

(^2) in ℝn, then



vv=+ 12 vv^222 ++ n, which is the distance from the origin to the associated point

P (v 1 , v 2 ,.. ., vn).

● (^) Vector representation of a Complex Number The vector representation of a complex
number z is the position vector



z associated to the point z in the complex plane.

If za=+bi for two real numbers a and b, then



z

a

b

=

é

ë

ê

ù

û

ú.