e sin φ =
|
| |
|
bb n
n
⋅
rr r
r
 or φ = 
sin–1
b n
b n
⋅
Example 25 Find the angle between the line
1
2
x +
 =
3
3
6
y
z −
=
and the plane 10 x + 2y – 11 z = 3 Solution Let θ be the angle between the line and the normal to the plane Converting the
given equations into vector form, we have
rr  =
ˆ
ˆ
ˆ
ˆ
ˆ
( –
3 )
( 2
3
6
)
i
k
i
j
k
+
+ λ
+
+
and
ˆ
ˆ
ˆ
( 10
2
11
)
r
i
j
k
⋅
+
−
r
 = 3
Here
b
r
 =
ˆ
ˆ
ˆ
2
3
6
i
j
k
+
+
   and  
k
j
i
n
11ˆ
2ˆ
10ˆ
−
+
=
r
sin φ =
2
2
2
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(2
3
6
) (10
2
11 )
2
3
6
10
2
11
i
j
k
i
j
k
+
+
⋅
+
−
+
+
+
+
=
40
7
15
−
×
 = 
−218
 = 8
21   or φ = 
1
8
sin
21
− 





Fig 11