15 find the vector equation of the line of

15. Find the vector equation of the line of intersection of the following two planes
t

⃗ w =
(
− 1 , − 1,0
)
+ t (− 1,2,1 )
16. Determine whether the following system of equations has a single point of intersection.
u
w

14 x − 27 z − 5 = 0 [ 7 ]
[ 5 ] × 2
14 x − 42 z = 0 [ 8 ]
[
8
]
−
[
7
]
− 15 z + 5 = 0
− 15 z =− 5
z =
5
15
z =
1
3
Sub
z =
1
3
into [5]
7 x − 21 z = 0
7 x − 21 (
1
3
)= 0
7 x − 7 = 0
7 x = 7
x = 1
Sub
x = 1 ∧ z =
1
3
into [1]
4 x + y − 9 z = 0
4 ( 1 )+ y − 9 (
1
3
)= 0
4 + y − 3 = 0
y =− 1
The point of intersection is
( 1 , − 1,
1
3
)
17. Find the shortest distance from
P (− 4,2,6 )
to the plane
2 x − 3 y + z − 8 = 0
⃗
n =( A,B ,C )
⃗
n =( 2 , − 3,1 )
Let
x = 0 ∧ y = 1
2 x − 3 y + z − 8 = 0
2
(
0
)
− 3
(
1
)
+ z − 8 = 0
z − 11 = 0
z = 11

Another point on the plane is
Q ( 0,1,11 )
⃗
PQ = Q − P
⃗
PQ =
(
− 4,2,6
)
−( 0,1,11 )
⃗
PQ =(− 4,1 , − 5 )
|
pro j
⃗ n
(
⃗
PQ
)
|
=
|
⃗
PQ ∙ ⃗ n
⃗ n∙ ⃗ n
|
∨ ⃗ n ∨ ¿
|
pro j
⃗ n
( ⃗
PQ
)
|
=
|
( − 4,1 , − 5 ) ∙ ( 2 , − 3,1 )
(
2 , − 3,1
)
∙
(
2 , − 3,1
)
|
∨( 2 , − 3,1 )∨ ¿
|
pro j
⃗ n
( ⃗
PQ
)
|
=
|
( − 8 − 3 − 5 )
( 4 + 9 + 1 )
|
∨
√
4 + 9 + 1 ∨ ¿
|
pro j
⃗ n
(
⃗
PQ
)
|
=
|
− 16
14
|
√
14
|
pro j
⃗ n
(
⃗
PQ
)
|
=
− 16
√
14
14
|
pro j
⃗ n
(
⃗
PQ
)
|
=− 4.28
(two decimal places)
The shortest distance between
P
and the plane is 4.28 units (to two decimal
places)