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23 Pages
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Department
Mathematics
Course Code
MAT237Y1
Professor
Stanzeck

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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
www.notesolution.com

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Description
MAT237Y1a.doc The Geometry of Euclidean Space N V ECTORS IN R n u =(u ,,u ) A vector in R can be written as 1 n . Definitions u 1= v1 u = v u = u1,,u n) v =(v1,,v n) un = vn If and , then . u = u1, un), R . Some Properties n u R n n u + v R 1) v R . R n u R nu R 2) . 3) u+ v = v+u . 4) u+ (v+w )= v+u +w . u = (u , ,u ) u+ (u)= 0 =(0, ,0)=(u )+u 5) 1 n and . 6) u 0 u = 0 u . 7) u+ v )=u+v . ( + u =u+ u 8) . ( u = (u) 9) . 10)1u =u . N ORM Definition 2 2 Let u = u1 u n). The norm isu = u 1 ++u n . Properties u 0 1) . u+ v u + v 2) . u = u 3) . Page 1 of 23 www.notesolution.com MAT237Y1a.doc D OT P RODUCT Let u = u1,,u n )and v = v1,,v n), thenuv = u1 1++u v n n. Properties 1) uv = vu. 2) u+ v )w = uw + vw . (uv)= u(v ) 3) . 4) uu 0 , moreover,uu = 0 u = 0 More Properties 2 1) uu = u . 2) uv u v (Cauchy-Schwartz). 2 2 2 2 3) u+ v u+(v = u ) +2uv+ v and (u v u( v = u) 2uv+ v . u+ v u(v = u )2 v 2 4) . The Cosine Law 2 2 2 v = v + u 2 v u cos 2 2 (vu )vu = v) + u 2 v u cos v 2 2 2 2 v - u v 2u + u = v + u 2 v u cos = v u cos u Theorem uv = v u cos,u,v 0 . Some Basic Consequences 1) uv u v . 0 < 2) uv > 0means 2 . = uv =0 means 2 . uv < 0means . P ROJECTION Page 2 of 23 www.notesolution.com
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