School

Pennsylvania State University **We aren't endorsed by this school

Course

MATH 220

Subject

Mathematics

Date

Oct 16, 2023

Pages

12

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Euclidean Spaces
Def: Let β be the set of all real numbers. We regard β
geometrically as the
Euclidean line
.
Ex:
Plot β2, 1.5, and π on the Euclidean line.
Def: Each ordered pair (π₯, π¦) of real numbers represents a point
in the π₯π¦ βplane. The set of all such ordered pairs is
Euclidean
2-space,
which is denoted β
ΰ¬Ά
, and often called
the plane
.
The axes meet at the point (0,0) which is called the
origin
.
Note: alternatively, we may use π₯
ΰ¬΅
for the horizontal axis, π₯
ΰ¬Ά
for
the vertical axis, and call β
ΰ¬Ά
the π₯
ΰ¬΅
π₯
ΰ¬Ά
-plane.
Vectors in β
ΰ¬Ά
Def: A pair of real numbers π and π produces a
two dimensional
vector
π£
β.
We write π£
β as a column and visualize π£
β as an arrow in β
ΰ¬Ά
with
tail located at the origin and head at the point (π, π).
Lecture 7: Euclidean Spaces
Math 220 Fall 2023 Lecture Notes Page 1

π is called the
first component
of π£
β
and π is the
second
component
.
Ex:
Sketch π£
β = ΰ΅€
β1
3
ΰ΅¨ and
π€
α¬α¬β = α
2
β4
α in β
ΰ¬Ά
.
Def: The
zero vector
0
α¬β
has components π = 0 and π = 0.
Def: Two vectors are
equal
if their corresponding entries are
equal.
Ex:
Determine if each pair of vectors are equal vectors.
(1) α
4
7
α and α
7
4
α
(2) ΰ΅€
1
5
ΰ΅¨ and ΰ΅€
4 β 3
4 + 1
ΰ΅¨
Math 220 Fall 2023 Lecture Notes Page 2

Vector Operations
Vector addition:
we add two vectors by adding component-wise.
If π£
β = ΰ΅€
π£
ΰ¬΅
π£
ΰ¬Ά
ΰ΅¨
and
π€
α¬α¬β = α
π€
ΰ¬΅
π€
ΰ¬Ά
α ,
π£
β + π€
α¬α¬β =
Geometrically, we add vectors using the
Parallelogram Law
:
v
α¬β + π€
α¬α¬β lies along the diagonal of the parallelogram with sides π£
β
and π€
α¬α¬β.
Ex:
Let π£
β = α
3
1
α and π€
α¬α¬β = α
β1
2
α. Find π£
β + π€
α¬α¬β geometrically using
the Parallelogram Law. Then confirm your answer algebraically.
Scalar multiplication
: let π be a
scalar
(real number) and π£
β be a
vector. The scalar multiple ππ£
β is the vector with components ππ£
ΰ¬΅
and ππ£
ΰ¬Ά
.
Ex:
Let π£
β = α
3
1
α . Find 2π£
β,
ΰ¬΅
ΰ¬Ά
β―π£
β, and βπ£
β. How does multiplication
by each scalar change the vector geometrically?
Math 220 Fall 2023 Lecture Notes Page 3

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