Eureka Math Algebra I Study Guide

terMInology | 117

● (^) 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.

● (^) Vector Scalar Multiplication For a vector



v in ℝn and a real number k, the scalar product

kv×



is the vector whose ith component is the product of k and the ith component of



v for

1 ££in. If k is a real number and





v

v

v

vn

=

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú

1

(^2) in ℝn, then kv
kv
kv
kvn

×=

é

ë

ê

ê

ê

ê

ù

û

ú

ú

ú

ú





1

(^2).
● (^) Vector Subtraction For vectors



v and



w, the difference



vw- is the sum of



v and the

opposite of



w; that is,



vw-=vw+-(). 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).
● (^) Zero Matrix The mn ́ zero matrix is the mn ́ matrix in which all entries are equal to
zero. For example, the 22 ́ zero matrix is

00

00

é

ë

ê

ù

û

ú, and the^3 ́^3 zero matrix is

000

000

000

é

ë

ê

ê

ê

ù

û

ú

ú

ú

.

● (^) Zero Vector The zero vector in ℝn is the vector in which each component is equal to zero.
For example, the zero vector in ℝ^2 is

0

0

é

ë

ê

ù

û

ú, and the zero vector in ℝ

(^3) is

0

0

0

é

ë

ê

ê

ê

ù

û

ú

ú

ú

.

Module 2

● (^) Argument The argument of the complex number z is the radian (or degree) measure
of the counterclockwise rotation of the complex plane about the origin that maps the
initial ray (i.e., the ray corresponding to the positive real axis) to the ray from the
origin through the complex number z in the complex plane. The argument of z is
denoted arg (z).
● (^) bound Vector A bound vector is a directed line segment (an arrow). For example, the
directed line segment AB



is a bound vector whose initial point (or tail) is A and terminal

point (or tip) is B.