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(b) (1) v ab = [−3, 0, 2 ]. v bc = [−3, 0, −4 ]. x bc z ab − x ab z bc = (−3)(2) − (−3)(−4) =
−18 < 0. Thus, the NPC is turning counterclockwise.
(2) v ab = [ 7, 0, 5 ]. v bc = [−1, 0, 3 ]. x bc z ab −x ab z bc = (−1)(5)−(7)(3) = −26 < 0.
Thus, the NPC is turning counterclockwise.
(3) v ab = [ 6, 0, −5 ]. v bc = [−12, 0, −5 ]. x bc z ab −x ab z bc = (−12)(−5)−(6)(−5) =
90 > 0. Thus, the NPC is turning clockwise.
(4) v ab = [ 3, 0, 1 ]. v bc = [ 3, 0, 2 ]. x bc z ab − x ab z bc = (3)(1) − (3)(2) = −3 < 0.
Thus, the NPC is turning counterclockwise.
23. p = p + (k −1) (pn) n
2
4
3
5
0
@
2
4
3
5
2
4
3
5
1
A
2
4
3
5
1
0
0
1
0
0
n x
n y
n z
n x
n y
n z
=
+ (k −1)
2
4
3
5
2
4
3
5
1
0
0
n x
n y
n z
=
+ (k −1) (n x )
2
3
2
3
(k −1) n x 2
(k −1) n x n y
(k −1) n x n z
1
0
0
4
5
4
5
=
+
2
4
3
5
1 + (k −1) n x 2
(k −1) n x n y
(k −1) n x n z
=
24. p = cos θ (p−(pn) n) + sin θ (n×p) + (pn) n
0
@
2
4
3
5
0
@
2
4
3
5
2
4
3
5
1
A
2
4
3
5
1
A
0
@
2
4
3
5
2
4
3
5
1
A
0
@
2
4
3
5
2
4
3
5
1
A
2
4
3
5
1
0
0
1
0
0
n x
n y
n z
n x
n y
n z
n x
n y
n z
1
0
0
1
0
0
n x
n y
n z
n x
n y
n z
= cos θ
+sin θ
×
+
0
@
2
4
3
5
2
4
3
5
1
A
2
4
3
5
2
4
3
5
1
0
0
n x
n y
n z
0
n z
−n y
n x
n y
n z
= cos θ
−n x
+ sin θ
+ n x
2
3
2
3
2
3
1−n x 2
−n x n y
−n x n z
n x 2
n x n y
n x n z
0
n z
−n y
4
5
4
5
4
5
= cos θ
+ sin θ
+
2
4
3
5
2
4
3
5
2
4
3
5
cos θ −n x 2 cos θ
−n x n y cos θ
−n x n z cos θ
n x 2
n x n y
n x n z
0
n z sin θ
−n y sin θ
=
+
+
2
4
3
5
cos θ −n x 2 cos θ + n x 2
−n x n y cos θ + n z sin θ + n x n y
−n x n z cos θ −n y sin θ + n x n z
=
2
4
3
5
n x 2 (1−cos θ) + cos θ
n x n y (1−cos θ) + n z sin θ
n x n z (1−cos θ)−n y sin θ
=
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