Exercises

Find the equation of the line through the given points.
1.
P1( 0,7) , P2( 1,2)
2.
P1( 0,3) , P2( 1,-3)
3.
P1( 7,9,2) , P2( 3,7,0)
4.
P1( 7,9,2) , P2( 3,7,0)
5.
P1( -4,-17,1) , P2( -3,3,5)
6.
P1( 0,0,0) , P2( 1,3,1)
7.
P1( p,e,2) , P2( p-e,p+e,0)
8.
P1( p,e-1,ln( 2) ) , P2(tan( 1) ,sin( 3) ,e)
  

Find the equation of the plane through the three given points.
9.
P1( 0,0,0) , P2( 1,2,1) , P3(2,1,1)
10.
P1( 0,0,0) , P2(2,3.2) , P3( 1,1,1)
11.
P1( 1,3,2) , P2( -2,5,7) ,  P3( 2,1,4)
12.
P1( -1,4,3) , P2( 3,4,6) , P3( 0,-3,2)
13.
P1( 0,0,0) ,  P2( 1,2,0) ,  P3( 2,1,0)
14.
P1( 1,3,2) , P2( 2,-7,9) , P3( -2,1,5)
15.
P1( 0,0,0) ,  P2( 1,1,0) ,  P3( 1,1,1)
16.
P1( 0,0,0) ,  P2( 0,0,1) ,  P3( 1,1,1)
17.
P1( 1,2,3) ,  P2( 0,-3,3) ,  P3( -5,-2,3)
18.
P1( 1,0,0) ,  P2( 0,1,10) ,  P3( 0,01)
 

Find the equation of the plane with the given description.
19.
Through ( 0,0,0) and L( t) =  ( 2t,3t,4)
20.
Through ( 1,3,2)  and L( t) = (2t-1,1-4t,7-2t)
21.
Through K( s) = ( s,0,0) and L( t) = ( 0,t,0)
22.
Through K( s) = ( s,0,0) and L( t) = ( 0,t,0)
23.
Through ( 0,0,0) and spanned by u = á2,1,2 ñ , v = á -2,5,4 ñ
24.
Through ( 2,1,3) and spanned by u = i-j, v = i+k
  
Verify the triple vector product for the following triples of vectors.
25.
i, j, k
26.
u = i+j, v = i-j,  w = k
27.
u = á 1,3,2 ñ , v = á 2,1,3 ñ ,  w = á-1,3,4 ñ
28.
u = aj+bk, v = ak-bj,  w = i

      

29. The points P1( 1,3,2) , P2(3,7,5) , and P3( 5,11,8) all lie on a straight line. Find the equation of the line through P1 and P2 and determine the value of the parameter corresponding to P3. Repeat using P1 and P3 to determine the line and find the value of the parameter corresponding to P2.

30.  Show that the three points P1(1,3,2) , P2( 3,7,5) , and P3( 5,11,8) all lie on a straight line. (i.e. vectors formed are all parallel). Then attempt to find the equation of the plane through the three points. What happens?

31. Does the line through the points P1( 1,2,1) and P2( 3,3,0) intersect the line through the points P3( 0,0,1) and P4(2,7,2) ?

32.  Does the line through the points P1(2,-3,1) and P2( -3,1,3) intersect the line through the points P3( 2,1,-4) and P4( -1,-1,-1) ? 

33.  For what value of k does the line through the points P1( 2,1,1) and P2( -2,0,-1) intersect the line through the points P3( 3,5,4) and P4( 2,3,k)

34. Show that if the line through P1(x1, y1) and P2( x2, y2) does not intersect the line through P3( x3, y3) and P4( x4, y4) , then
P1P2
  is parallel to  
P3P4

35. Are the points P1( 0,1,2) , P2( 3,2,5) , P3( 1,3,7) , and P4(5,1,3) all in the same plane? Explain.

36. Are the points P1( 0,2,4) , P2( 2,5,1) , P3( 1,1,4) , and P4(2,7,5) all in the same plane? Explain.

37. A baseball thrown from a height of 6 feet drops 5 feet and curves 0.5 feet to the left once it reaches the plate 66.5 feet away.
Assuming motion of the ball is in a plane, what plane contains the trajectory of the ball?  

38. What would the equation of the plane in exercise 37 be if the ball broke to the right instead of to the left?

39.  Suppose that u is orthogonal to a unit vector n and that  u^ = n×u.  Show that
uq = cos( q) u+sin(q) u^
is a vector in the plane spanned by u and u^ that is the same length as u and u^ and that forms an angle of q with u.

40.  Write to Learn: Write an essay in which you use the triple scalar product and the triple vector product to show that
( a×b) ·( c×d) = ( a·c) ( b·d) -( a·d) ( b·c)

      

Exercises 41-44 explore interpretations and applications of the triple vector product.

41.  Use the triple vector product to show that cross product multiplication satisfies
( u×v) ×w+( w×u) ×v+( v×w) ×u = 0

42.  Suppose that P is a plane with normal vector n and containing point P( x1,y1,z1) and suppose that n is a unit vector.  Then the projection of a vector w into P is defined to be
projP( w) = n×( w×n)
Show that if w is parallel to P, which is to say that the endpoint of w is in the plane if the initial point of w is at P( x1,y1,z1) , then
projP( w) = w

43.  Write to Learn:  Let u be the direction vector of a non-zero vector p, and let v be a vector.  Show that
projp( v) = v-u×( v×u)
Use the result to explain the significance of the vector u×( v×u) when u is a unit vector.  

44.  Write to Learn:  Suppose a planet is located at the tip of a vector r at time t as it orbits a sun located at the origin of a 3-dimensional coordinate system.
 The acceleration of the planet about the sun is given by the inverse square law
a =  -GM
r2
 u
where r = ||r|| is the distance of the planet from the sun, M is the mass of the sun, G is the universal gravitational constant, and u is the direction vector of r.   Assuming that the planet's orbit is in the xy-plane, the angular velocity vector is given by
L = r2  dq
dt
 ( u×u^)
where q is the angle between r and i (i.e., the x-axis) at time t and where u^ is the unit vector in the xy-plane that forms a 90° angle with u at time t.  Calculate a×L and explain the significance of the result, such as the direction that a×L is pointing in and what the magnitude of a×L would represent if GM = 1.