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Department
Mathematics
Course Code
MAT237Y1
Professor
Sean Uppal

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MAT237Y1a.doc
Page 1 of 23
The Geometry of Euclidean Space
VECTORS IN RN
A vector in Rn can be written as
(
)
n
uu ,,
1=u.
Definitions
If
(
)
n
uu ,,
1=u and
(
)
n
vv ,,
1=v, then
=
=
=
nn vu
vu
11
vu
.
(
)
Ru=
αααα
,,
1n
uu .
Some Properties
1)
( )
n
n
n
Rvu
Rv
Ru+
.
2)
n
nRu
Ru
R
α
α
.
3)
u
v
v
u
+
=
+
.
4)
(
)
(
)
wuvwvu++=++ .
5)
(
)
n
uu =,,
1u and
(
)
(
)
(
)
uu0uu +===+ 0,,0.
6) u0u0u
+
=
=
+
.
7)
(
)
vuvu
ααα
+=+ .
8)
(
)
uuu
βαβα
+=+ .
9)
(
)
(
)
uu
βααβ
=.
10) uu
=
1.
NORM
Definition
Let
(
)
n
uu ,,
1=u. The norm is
22
1n
uu ++= u.
Properties
1) 0u.
2) vuvu++ .
3) uu
αα
=.
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MAT237Y1a.doc
Page 2 of 23
DOT PRODUCT
Let
(
)
n
uu ,,
1=u and
(
)
n
vv ,,
1=v, then nn vuvu++=
11
vu.
Properties
1)
u
v
v
u
=
.
2)
(
)
wvwuwvu+=+.
3)
(
)
(
)
vuvu
αα
=.
4) 0
uu , moreover, 0uuu
=
=
0
More Properties
1)
2
uuu =.
2) vuvu (Cauchy-Schwartz).
3)
( ) ( )
22 2vvuuvuvu++=++ and
( ) ( )
22 2vvuuvuvu+=.
4)
( ) ( )
22 vuvuvu=+.
The Cosine Law
( ) ( )
θ
θ
θ
θ
cos
cos22
cos2
cos2
2222
22
222
uvvu
uvuvuvuv
uvuvuvuv
uvuvuv
=
+=+
+=
+=
Theorem
0vuuvvu=,,cos
θ
.
Some Basic Consequences
1) vuvu.
2) 0
>
vu means
2
0
π
θ
<
.
0
=
vu means
2
π
θ
=
.
0
<
vu means .
PROJECTION
θ
u
v
v
-
u
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MAT237Y1a.doc
Page 3 of 23
Geometry
Theorem
1) u
vu
v
u
=proj
.
2)
u
uu
vu
v
u
=proj
.
LINES IN R3
Some ways to characterize the line that passes through a given point
(
)
0000 ,, zyxP= and follows the
direction of a non-zero vector
(
)
cba ,,=v. If
(
)
zyxP,,= denotes any point on the line, then:
v
λ
=PP0 (vector equation).
czz
byy
axx
λ
λ
λ
+=
+=
+
=
0
0
0
(parametric equations).
If 0,, cba , c
zz
b
yy
a
xx 000
`
=
=
(symmetric equations).
THE CROSS PRODUCT
Definition
If
(
)
321 ,, uuu=u and
(
)
321 ,, vvv=v, then
(
)
122131132332 ,, vuvuvuvuvuvu=× vu is the cross
product of u and v.
u
v
v
u
proj
(
)
0000 ,, zyxP=
(
)
zyxP,,=
(
)
cba ,,=v
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