MATH1131 Lecture Notes - Lecture 6: Cross Product, Lunar Reconnaissance Orbiter, Lithium Chloride
3 Jun 2018
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2.2 Angle between two vectors.
Example 2.5. LeL A, B be points and let 0 be the
f
angle between a: OA and b : OB. By applying the
cosine rule AOAB prove that
Definition 2.6. Lel, a, b e 1Rn. fhe angle 0 between
a and,b is gi,uen by
4
-2
1
2
4
0,
(
{=
=J
=l
l.r
=
)
rrt
lgt
t!t
h =h,
\s
ft) ant r=
(-i) = -
Ii)
(,);
. Jii-
=)T
\Me now use the above result to come up with the defini-
tion of the a,rgle between two vectors in arbitrary dimen-
sions, iust as we did for length.
Example 2.8. If the angie between a and b is 0, What
is the angle between a and -b? What is the angle be-
Su, Cr"f O =
J@AsetS
orthonormal'i/
r.k
r4ilt'i
ros { (
o
0=
2.3 Orthogonality and projection
Recall that given vectors a, b € IR" the a.ngle d between
the fivo vecton and the dot product of the two vecto:s
are related Lry
a.b:lallblcos6.
If we substitur" Qffrhus we have the
following definition:
Definition 2.9. Leta,t E lR."
" th anqb hlwrtn
fir aJ6h b*unn
(. ,ad -l ir
sL tnl L 't(
E-0
0
-
4 ( 'niiI= ltl' * l!l' - > l{'llf tor 9
0 Jriltr fA =
tfd'
t{rort , tll"+ t},1'- )g*
r,I :14,1. l&l ro,r
b- e
I t i-ct.(r-r)
= lgl. - b,!-f L;r- lfl'
- lIl' + lgl'- r {..!
: li'cl'-- lrl'l ll.l" -rEl'lU ol 0
e tfoyrn) r
vi vj : {?' ',!',
Let's see what this looks like with Geocebra.
'Th h$4rrd buit t^ n3
(;) , (q) ,(i)
=# rn)
#) : rs-6r s,
, vo) is sa'id, to be
m whereo4o.r.
Example 2.7. Find the angle between
tlrn
=5
r2+O + 6 =la \,
,r*l'+0rtS'
