11.5 Equations of straight Lines and Planes


1-7. Find a vector equation, parametric equations and symmetric equations for the line.


1. Through the point and parallel to the vector .

Sol) 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

Sol)

 

var('t');

r0=vector([3,5,1]);

pv=vector([4,1,-1]);

r0+t*pv

  (4*t + 3, t + 5, -t + 1)


       parametric equation: , , ,

       symmetric equation: .


3. Through the origin and parallel to the line , , .

 Sol) This line has the same direction as the given line, .

     Here , so a vector equation is and parametric equations are

     . The symmetric equations are .


4. Through the point and perpendicular to the plane .

 Sol) vector equation: ,

     parametric equation: , , ,

     symmetric equation: .


5. Through the origin and the point .

Sol) 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

 Sol) parametric equation: , , ,

      symmetric equation: .

 

var('t');

A=vector([3,5,-3]);

B=vector([-1,0,5]);

v=B-A

A+t*v

 

  (-4*t + 3, -5*t + 5, 8*t - 3)




7. Through and perpendicular to both and .

 Sol) 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

 Sol) The lines are not parallel because the corresponding vectors , are not parallel.

 

var('t,s');

A=vector([3,4,5]);

B=vector([-2,0,1]);

v=B-A

print A+t*v

C=vector([2,1,4]);

D=vector([-3,-3,-3]);

w=D-C

print C+s*w

  (-5*t + 3, -4*t + 4, -4*t + 5)

  (-5*s + 2, -4*s + 1, -7*s + 4)


9.Is the line through and perpendicular to the line through and ?

 Sol) 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

 Sol) skew lines

var('t,s');

L1=parametric_plot3d((25,1+3*t,-4*t),(t,-50,50))

L2=parametric_plot3d((3+s,4-2*s,s),(s,-50,50),color="red")

show(L1+L2)


 


11.: , , ,

    : , , .


 Sol) Since the direction vectors are and ,

      we have . Hence the lines are parallel.


12. : ,

    : .

 Sol)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

        ,

        ,

       ,

      .

, are intersecting.


13. : ,

          : .

http://matrix.skku.ac.kr/cal-lab/cal-11-5-13.html

 Sol)

 

s,t=var('t,s')

a=(3,1,4)

b=(-2,2,1)

at=(4,3,7)

bt=(3,0,2)

A=parametric_plot3d( (4+3*t, 3+t, 7+4*t), (t, -30, 30))

B=parametric_plot3d( (3-2*s, 2*s, 2+s), (s, -30, 30),color="red")

show(A+B) 


             


 

Ab=matrix(QQ,3,3,[a[0],-b[0],bt[0]-at[0],a[1],-b[1],bt[1]-at[1],a[2],-b[2],bt[2]-at[2]])

Ab.echelon_form()

  [ 1  0 -1]

  [ 0  1  1]

  [ 0  0  0]


 

t= -1

s=1

A=(4+3*t, 3+t, 7+4*t)

B=(3-2*s, 2*s, 2+s)

A,B 

  ((1, 2, 3), (1, 2, 3))


14-15. Find an equation of the plane.


14. Through the point and perpendicular to the vector .

 Sol)

     .


15.Through the point and with normal vector

 Sol) 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?

   ,   ,

    .

 Sol) and are parallel.

 

17. Which of the following four lines are parallel?

    ,    ,

    ,             .

 Sol) 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. , , .

 Sol) is a normal vector to the plane and is a point of the plane.

     Then or to be the equation of the plane.


19. , , .

 Sol) 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

 Sol)

 

a=vector(QQ, [2, -1, 3])

d= -4

p=vector(QQ, [3, 1, 5])

dis=abs(a.dot_product(p)+d)/a.norm()

dis 

  8/7*sqrt(14)


21. .

 Sol) The distance .


22-23. Find the distance between the given parallel planes.


22. .

 Sol) : , : .

      In particular, we put a point in first plane. Then, the distance between and the plane is .


23. .

 Sol) Put in the equation of the first plane to get the point on the plane.

     Since the planes are parallel, the distance between them 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

    .