Calculus-Sec-11-5-Solution
11.5 Equations of straight Lines and Planes
by SGLee - HSKim - JHLee
1-7. Find a vector equation, parametric equations and symmetric equations for the line.
1. Through the point 
and parallel to the vector 
.
For this line, we have 
and 
. Hence a vector equation is
and parametric equations are
.
The symmetric equations are 
.
2. Through the point 
and parallel to the vector 
.
http://matrix.skku.ac.kr/cal-lab/cal-11-5-2.html
(You may do it with Sage in http://math1.skku.ac.kr/ .
Open resources in http://math1.skku.ac.kr/pub/ )
parametric equation: 
, 
, 
,
symmetric equation: 
.
3. Through the origin and parallel to the line 
, 
, 
.
This line has the same direction as the vector, 
.
Here 
, so a vector equation is 
and parametric equations are
. The symmetric equations are 
.
4. Through the point 
and perpendicular to the plane 
.
Normal vector : (2, -1, -2)
(
ℝ) =>
vector equation: 
,
parametric equation: 
, 
, 
,
symmetric equation: 
.
5. Through the origin and the point 
.
For this line, we have 
and 
. Hence a vector equation is
and parametric equations are 
.
The symmetric equations are 
.
6. Through the points 
and 
.
http://matrix.skku.ac.kr/cal-lab/cal-11-5-6.html
parametric equation: 
, 
, 
,
symmetric equation: 
.
7. Through 
and perpendicular to both 
and 
.
A line perpendicular to the given two vectors has the same direction as a cross product of the two vectors. That is,
.
Here, 
, so a vector equation is 
and
parametric equations are 
.
The symmetric equations are 
.
8. Is the line through 
and 
parallel to the line through 
and 
?
http://matrix.skku.ac.kr/cal-lab/cal-11-5-8.html
The lines are not parallel because the corresponding vectors
, 
are not parallel.
9. Is the line through 
and 
perpendicular to the line through 
and 
?
Direction vectors of the lines are 
and 
.
Since 
, the vectors and the lines are not perpendicular.
10-13. Determine whether the lines 
and 
are parallel, skew, or intersecting. If they intersect, find the point of intersection.
10. 
: 
, 
, 
,
: 
, 
, 
.
http://matrix.skku.ac.kr/cal-lab/cal-11-5-10.html
It is apparent that the lines are skew in the following figure.
11. 
: 
, 
, 
, 
ℝ.
: 
, 
, 
, 
ℝ.
Since the direction vectors are 
and 
,
we have 
. Hence the lines are parallel.
12. 
: 
,
: 
.
The lines are not parallel because the corresponding vectors 
, 
are not parallel.
If 
and 
have a point of intersection, there would be values of 
and 
such that
,
,
. 
and 
are skew.
13. 
: 
,
: 
.
http://matrix.skku.ac.kr/cal-lab/cal-11-5-13.html
From the figure it is clear that the lines are intersecting.
14-15. Find an equation of the plane.
14. Through the point 
and perpendicular to the vector 
.
.
15.Through the point 
and with normal vector 
.
is a normal vector to the plane and 
is a point of the plane.
Then 
or 
to be the equation of the plane.
16. Which of the following four planes are parallel?
, 
,
, 
.
and 
are parallel.
17. Which of the following four lines are parallel?
, 
,
, 
.
and 
are parallel.
18-19. Find an equation of the plane through the given point with the normal vector
which is the direction of the line with the given parametric equations.
18. 
, 
, 
is a normal vector to the plane and 
is a point of the plane.
Then 
or 
to be the equation of the plane.
19. 
, 
, 
is a normal vector to the plane and 
is a point of the plane.
Then 
or 
is the equation of the plane.
20-21. Find the distance from the point to the given plane.
20. 
http://matrix.skku.ac.kr/cal-lab/cal-11-5-20.html
The normal vector to the plane is n= <2, -1, 3>
21. 
The distance 
.
22-23. Find the distance between the given parallel planes.
22. 
: 
, 
: 
. Note that (-1, 0, 1) is a point of the first plane.
Since the planes are parallel, the distance 
between the two planes is the distance from 
to the second plane.
Then, the distance between 
and the plane 
is
.
23. 
Put 
in the equation of the first plane to get the point 
on the plane.
Since the planes are parallel, the distance 
between the two planes is the distance from 
to the second plane.
Hence 
.
24. Find the distance between the two skew lines
and 
.
25. Prove that the distance between the parallel planes 
and 
is
.
26. (Line of intersection of two planes)
Plot the two planes 
and 
.
Find the line of intersection of two planes and hence plot this.
27. (Line of intersection of two planes)
If the two lines have a point in common then there exists 
and 
such that 
.
The above system must have a unique solution. Let is verify this and find a common point.
Clearly, 
gives a unique solution and point 
is the common point.
