(a) Let P be a point not on the plane that passes through the points Q , R , and S . Show that the distance d from P to the plane is d = | a ⋅ ( b × c ) | | a × b | where a = Q R → , b = Q S → , and c = Q P → . (b) Use the formula in part (a) to find the distance from the point P ( 2 , 1 , 4 ) to the plane through the points Q ( 1 , 0 , 0 ) , R ( 0 , 2 , 0 ) , and S ( 0 , 0 , 3 ) .
(a) Let P be a point not on the plane that passes through the points Q , R , and S . Show that the distance d from P to the plane is d = | a ⋅ ( b × c ) | | a × b | where a = Q R → , b = Q S → , and c = Q P → . (b) Use the formula in part (a) to find the distance from the point P ( 2 , 1 , 4 ) to the plane through the points Q ( 1 , 0 , 0 ) , R ( 0 , 2 , 0 ) , and S ( 0 , 0 , 3 ) .
Solution Summary: The author explains that the distance from the point P to the plane is the length of a perpendicular line. The volume of the parallelepiped defined by the vectors QR,QS and
(a) Let P be a point not on the plane that passes through the points Q, R, and S. Show that the distance d from P to the plane is
d
=
|
a
⋅
(
b
×
c
)
|
|
a
×
b
|
where
a
=
Q
R
→
,
b
=
Q
S
→
, and
c
=
Q
P
→
.
(b) Use the formula in part (a) to find the distance from the point
P
(
2
,
1
,
4
)
to the plane through the points
Q
(
1
,
0
,
0
)
,
R
(
0
,
2
,
0
)
, and
S
(
0
,
0
,
3
)
.
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