Game Development Reference
In-Depth Information
v
⊥
= v −v
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
4
−1
7/2
7/2
0
4−7/2
3−7/2
−1−0
1/2
−1/2
−1
=
−
=
=
11. Define a triangle using the vectors a, b, and a − b, and let θ be the angle between a and
b. Then the squared length of the edge a−b is:
a−b
2
= (a−b)(a−b)
= aa−2ab + bb
= aa + bb−2ab
= a
2
+b
2
−2abcos θ
which is the law of cosines.
12. First, let's obtain some information about the vector components.
From the figure, we have
a
x
= acos θ
a
, a
y
= asin θ
a
,
b
x
= bcos θ
b
, b
y
= bsin θ
b
.
Now we can proceed with the algebraic definition of the dot product and the cosine differ-
ence identity:
ab = a
x
b
x
+ a
y
b
y
= acos θ
a
bcos θ
b
+asin θ
a
bsin θ
b
= ab(cos θ
a
cos θ
b
+ sin θ
a
sin θ
b
)
= abcos (θ
b
−θ
a
)
= abcos θ.
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