(
b): Find the equation of the line perpendicular to the one in (a),
and passing through Q(1,4).
Solution:
We need vector
&
such that
&
. °
= 0,
with
°
=
1
2
.
()*
&
+
&
,
.
1
2
= 0
or
&
+
+ 2&
,
= 0.
Let
&
,
= −1,
then
&
+
= 2
,
or
&
=
2
−1
.
Then

=
.
/
= ¹ + 0&
=
1
4
+ 0
2
−1
.
1
9/2/2015
4
(
c): Find the coordinates of the point of intersection
between the two lines.
Solution:
We need point R.
2±¼3 ¸¿ = (., /) =
2,2 + 4(1,2)
2±¼3¹¿ =
., /
=
1,4 + 0(2, −1)
Now the xvalues and yvalues are the same at
intersection:
For BR:
. = 2 +
4
/ = 2 + 2
4
For QR:
. = 1 + 2
0
/ = 4 − 1
0
2 + 4 = 1 + 20
5±637
4 = −1 + 20
And
2 + 24 = 4 − 10
(1)
Substitute the
4
value in (1):
2 + 2
−1 + 20
= 4 − 0
2 − 2 + 40 = 4 − 0
50 = 4
0 =
8
9
= 0.8
But
4 = −1 + 20
4 = −1 + 2
0.8
4 = 0.6
From:
For BR:
. = 2 +
4
/ = 2 + 2
4
. = 2 + 0.6 = 2.6
°±;
/ = 2 + 2
0.6
= 3.2
For QR:
. = 1 + 2
0
/ = 4 − 1
0
. = 1 + 2
0.8
= 2.6
°±;
/ = 4 −
0.8
= 3.2
So:
² = (2.6, 3.2)
9/2/2015
5
EXAMPLE
Example 3
.
Find the line of intersection between the planes
. + / + < = 2
and
2. − < = 0
.
Solution:
Parameterize (Create a parameter):
Choose
. = 4,
(you can choose any of x, y or z)
Then
. = 4
< = 2. = 24
/ = 2 − . − < = 2 − 4 − 24 = 2 −
34
.
/
<
=
4
2 − 34
24
Take apart:
.
/
<
=
4
2 − 34
24
=
0 + 4
2 − 34
0 + 24
=
4
−34
24
+
0
2
0
=
0
2
0
+ 4
1
−3
2
,
=>

=
0
2
0
+ 4
1
−3
2
Vector equation of the line of intersection.
9/2/2015
6
NOW YOU:
Find the vector equation of the line of intersection
between the planes:
2
. − / + < = 2,
2. + / +
3< = 6.
Next find the angle between the planes.
ANOTHER ONE FOR YOU TO TRY…
Find the scalar equation of the plane which
contains 3 points P
0
, P
1
and P
2
at any position on
the plane.
Scalar equation:
°. + ?/ + &< = ;
@
A
=
4,5, −7
@
B
=
1,2, −3
@
±
= (0,1,0)
C
D
C
²
C
E
9/2/2015
7
MORE EXAMPLES
•
We can not handle all the examples in class with all the
different combinations of lines and planes.
•
Make sure to go through ALL the additional exercises that
are on Blackboard in connection with the lines and planes.
•
If you should have any problems, make sure to attend the
practical sessions on Tuesday (WWG113) or Wednesday
(W201) to ask for assistance OR attend the walkin hours.
•
It is YOUR responsibility to get enough exercise with this.
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 Summer '20
 Vector Space