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Unformatted text preview: ally, d is the length of the perpen−→
−
dicular P D from P to shown in the ﬁgure.
Now by right angle trigonometry, u × v = u v sin θ = 15 sin θ
where 0 ≤ θ ≤ π is the angle between u
and v. Now j lies in the xy plane, so the
angle θ between j and v varies from 0 to π .
Consequently, u × v has
maximum length = 15
009 . d = b sin θ .
On the other hand,
a × b = a b sin θ ;
i.e.,
b sin θ = 10.0 points Let be the line passing through points
Q, R and P a point not on as shown in 4 a × b
.
 a But then
d= a × b
.
 a R
keywords: distance, distance from line, cross
product, vectors D
a
d
Q 010 θ
b P Express the distance d from P to in terms of
the vectors
−
−
→
−
−
→
a = QR ,
b = QP . 10.0 points When P is a point not on the plane passing
through the points Q, R, and S , then the
distance, d, from P to that plane is given by
the formula
d=  a · ( b × c) 
a × b cai (atc667) – HW21  Sect. 12.4 – chavezdominguez – (55235)
where 5 Consequently, P is at a −
−
→
a = QR , −
→
b = QS , −
−
→
c = QP . Use this formula to determine the distance
from P (2 , 0, −2) to the plane through the
points
Q(2, −2, 1), R(1, −2, 2), S (3, 0, 1) .
1. distance = 3
2 2. distance = 4
3. distance = 5
3 4. distance = 9
2 5. distance = 8
correct
3 6. distance = 3
Explanation:
For the given points,
−
−
→
a = QR = i + 2 j ,
while
−
→
b = QS = − i + k ,
and
−
−
→
c = QP = 2 j − 3 k .
In this case
a×b = i
1
−1 jk
20
01 = 2i− j + 2k, in which case
a × b = 22 + 1 + 22 = 3 . On the other hand,
a · ( b × c) = 1
−1
0 2
0
2 0
1
−3 = −8 . distance = 8
−8
=
3
3 from the plane through Q, R and S .
keywords: distance from plane, distance,
plane, scalar triple product, cross product,...
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 Fall '07
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