Eureka Math Algebra I Study Guide

118 | eureka Math algebra I Study guIde

Bound vectors are bound to a particular location in space. A bound vector AB

has a

magnitude given by the length of AB and direction given by the ray AB

. Many times,
only the magnitude and direction of a bound vector matters, not its position in space.
In that case, any translation of that bound vector is considered to represent the same
free vector.

● (^) Complex Number A complex number is a number that can be represented by a point in
the complex plane. A complex number can be expressed in two forms:

1. 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. Note
   that both the real and imaginary parts of a complex number are themselves real
   numbers.


2. For z¹^0 , the polar form of a complex number z is ri(cos()qq+ sin)() where rz= and
   q=arg(z), and i is the imaginary unit.

● (^) Complex Plane The complex plane is a Cartesian plane equipped with addition and
multiplication operators defined on ordered pairs by the following:
○ (^) Addition: ()ab,,+=()cd (,ac++bd).
When expressed in rectangular form, if za=+bi and w=+cdi, then
zw+=()ac++()bd+ i.
Multiplication: ()ab,,×=()cd (,ac-+bdad bc).
When expressed in rectangular form, if za=+bi and w=+cdi, then
zw×=()ac-+bd ()ad+bci. The horizontal axis corresponding to points of the form
(x, 0) is called the real axis, and a vertical axis corresponding to points of the
form (0, y) is called the imaginary axis.
● (^) Conjugate The conjugate of a complex number of the form ab+ i is ab- i. The conjugate
of z is denoted z.
● (^) Determinant of 22 ́ ́ Matrix The determinant of the 22 ́ matrix
ab
cd
é
ë
ê
ù
û
ú is the number
computed by evaluating ad-bc and is denoted by det
ab
cd
é
ë
ê
ù
û
ú
æ
è
ç
ö
ø

÷.

● (^) Determinant of 33 ́ ́ Matrix The determinant of the 3 ́ 3 matrix
aaa
aaa
aaa
11 12 13
21 22 23
31 32 33
é
ë
ê
ê
ê
ù
û
ú
ú
ú
is the
number computed by evaluating the expression,
a
aa
aa
a
aa
(^11) aa
22 23
32 33
12
21 23
31 33
detdet
é
ë
ê
ù
û
ú
æ
è
ç
ö
ø

÷-

é

ë

ê

ù

û

ú

æ

è

ç

öö

ø

÷+

é

ë

ê

ù

û

ú

æ

è

ç

ö

ø

a ÷

aa

(^13) aa
21 22
31 32
det ,
and is denoted by det
aaa
aaa
aaa
11 12 13
21 22 23
31 32 33
é
ë
ê
ê
ê
ù
û
ú
ú
ú
æ
è
ç
ç
ç
ö
ø

÷

÷

÷

.

● (^) Directed graph A directed graph is an ordered pair D (V, E) with
○ (^) V a set whose elements are called vertices or nodes, and
○ (^) E a set of ordered pairs of vertices, called arcs or directed edges.