Game Development Reference
In-Depth Information
PQ
×=
PQ
sin α
.
Gram-Schmidt Orthogonalization
A basis
for an n -dimensional vector space can be orthogo-
nalized by constructing a new set of vectors
=
ee
,
,
,
e
{
}
12
n
=
ee
,
,
,
e
using the formula
{
}
12
n
ee
i
1
i
k
e
=−
e
e
.
i
i
k
e
2
k
k
=
1
Exercises for Chapter 2
1. Let
P
=
2, 2,1
and
=−
1,
2, 0
. Calculate the following.
Q
(a)
P Q
(b)
PQ
×
(c) proj P Q
2.
Orthogonalize the following set of vectors.
e
e
e
=
=− −
=−−
22
,
, 0
1
22
1, 1,
1
2
0,
2,
2
3
3.
Calculate the area of the triangle whose vertices lie at the points 1, 2, 3 ,
2, 2, 4
, and 7,
8, 6
.
(
)
2
2
2
2
4. Show that
VW
+×=
V W
VW
for any two vectors V and W .
5.
Prove that for any three 3D vectors P , Q , and R ,
(
)
(
)
PQR PRQ QRP .
×× = ⋅
− ⋅
6.
Prove that for any two vectors P and Q ,
PQ
−≥ −
P Q ,
and show that this implies the extended triangle inequality,
PQPQPQ .
−≤+≤+
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