Eureka Math Algebra I Study Guide

122 | eureka Math algebra I Study guIde

● (^) 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 3

● (^) ellipse An ellipse is the set of all points in a plane such that the sum of the distances from
two points (foci) to any point on the line is constant. Given k, foci A and B, and any point
P on the ellipse, PA+=PB k.
Standard equation for an ellipse: xa by
2
2
2
+= 2 1
An ellipse whose foci points are the same point, that is, AB= , is a circle.
● (^) Horizontal Asymptote Let L be a real number. The line given by the equation yL= is a
horizontal asymptote of the graph of yf= ()x if at least one of the following statements
is true.
○ (^) As x approaches infinity, f(x) approaches L.
○ (^) As x approaches negative infinity, f(x) approaches L.
● (^) Hyperbola A hyperbola is the set of points in a plane whose distances to two fixed points
A and B, called the foci, have a constant difference. Given P and a positive constant, k,
PA-=PB k.
Standard equation for a hyperbola: xa yb
2
2
2
-= 2 1 or
y
a
x
b
2
2
2
-= 2 1
● (^) Vertical Asymptote Let a be a real number. The line given by the equation xa= is a
vertical asymptote of the graph of yf= ()x if at least one of the following statements
is true.
○ (^) As x approaches a, f(x) approaches infinity.
○ (^) As x approaches a, f(x) approaches negative infinity.