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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