Find the vector equation of the line of intersection for each pair of planes.
\displaystyle
\begin{array}{llllllll}
&x + 5y - 3z - 8 = 0 \\
&y + 2z - 4 = 0
\end{array}
Find the vector equation of the line of intersection for each pair of planes.
\displaystyle
\begin{array}{llllllll}
&5x - y +z - 22= 0\\
&x + 3y - 9=0
\end{array}
Find the vector equation of the line of intersection for each pair of planes.
\displaystyle
\begin{array}{llllllll}
&2x - y+z-22=0\\
&x -11y+2z -8=0
\end{array}
Find the vector equation of the line of intersection for each pair of planes.
\displaystyle
\begin{array}{llllllll}
&3x + y - 5z - 7=0\\
&2x - y -21z + 33 = 0
\end{array}
Which pairs of planes are parallel and distinct and which are coincident?
\displaystyle
\begin{array}{llllllll}
&2x +3y - 7z -2= 0\\
&4x + 6y -14z -8=0
\end{array}
Which pairs of planes are parallel and distinct and which are coincident?
\displaystyle
\begin{array}{llllllll}
&3x + 9y - 6z -24 = 0\\
&4x + 12y -8z - 32= 0
\end{array}
Which pairs of planes are parallel and distinct and which are coincident?
\displaystyle
\begin{array}{llllllll}
&4x-12y-16z-52 =0\\
&-6x +18y+24z+78=0
\end{array}
Which pairs of planes are parallel and distinct and which are coincident?
\displaystyle
\begin{array}{llllllll}
&x - 2y+2.5z -1=0\\
&3x -6y+7.5z -3= 0
\end{array}
Show that the line (x,y,z)=[10, 5, 16]+t[3, 1,5]
is contained in each of these planes.
\displaystyle
x + 2y -z-4= 0
Show that the line (x,y,z)=[10, 5, 16]+t[3, 1,5]
is contained in each of these planes.
\displaystyle
9x -2y-5z= 0
For each system of equations, determine the point of intersection.
\displaystyle
\begin{array}{llllllll}
&x + y +z-7=0\\
&2x + y+3z-17=0\\
&2x -y-2z+5=0
\end{array}
For each system of equations, determine the point of intersection.
\displaystyle
\begin{array}{llllllll}
&2x + y + 4z = 15\\
&2x + 3y + z = -6\\
&2x -y + 2z = 12
\end{array}
For each system of equations, determine the point of intersection.
\displaystyle
\begin{array}{llllllll}
&5x -2y - 7z -19=0\\
&x - y + z -8=0\\
&3x + 4y+ z - 1=0
\end{array}
For each system of equations, determine the point of intersection.
\displaystyle
\begin{array}{llllllll}
&2x-5y-z = 9\\
&x + 2y + 2z = -13\\
&2x + 8y + 3z = - 19
\end{array}
Determine the line of intersection of each system of equations.
\displaystyle
\begin{array}{llllllll}
&2x+y+z= 7\\
&4x +3y-3z=13\\
&4x+2y+2z=14
\end{array}
Determine the line of intersection of each system of equations.
\displaystyle
\begin{array}{llllllll}
&x + 3y-z=4\\
&3x+8y-4z= 4\\
&x +2y-2z= -4
\end{array}
Determine the line of intersection of each system of equations.
\displaystyle
\begin{array}{llllllll}
&x + 9y+3z= 23\\
&x + 15y + 3z= 29\\
&4x -13y + 12z = 43
\end{array}
Determine the line of intersection of each system of equations.
\displaystyle
\begin{array}{llllllll}
&x-6y+ z= -1\\
&x - y = 5\\
&2x-12y+ 2z= -2
\end{array}
In the system, at least one pair of planes are parallel. Describe each system.
\displaystyle
\begin{array}{llllllll}
&3x+15y-9z=12\\
&6x+30y-18z=24\\
&5x+25y -15z =10
\end{array}
In the system, at least one pair of planes are parallel. Describe each system.
\displaystyle
\begin{array}{llllllll}
&2x-y+4z= 5\\
&6x-3y+12z =15\\
&4x -2y + 8z =10
\end{array}
In the system, at least one pair of planes are parallel. Describe each system.
\displaystyle
\begin{array}{llllllll}
&3x+2y-z=8\\
&12x + 8y-4z =20\\
&18x+12y -6z = -3
\end{array}
In the system, at least one pair of planes are parallel. Describe each system.
\displaystyle
\begin{array}{llllllll}
&8x+4y+6z= 7\\
&12x + 6y + 9z = 1\\
&3x+ 2y +4z = 1
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&3x+y-2z= 18\\
&6x -4y + 10z = -10\\
&3x -5y+10z =10
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&2x+ 5y-3z= 12\\
&3x-2y+3z=5\\
&4x + 10y-6z= -10
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&2x-3y+2z= 10\\
&5x-15y + 5z = 25\\
&-4x + 6y -4z = -4
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&3x-5y-5z = 1\\
&-6x+ 10y + 10z = -2\\
&15x-25y -25z = 20
\end{array}
Planes that are perpendicular to one another are said to be orthogonal . Determine which of the systems from question 4 contain orthogonal planes.
For each system of planes below, find the triple scalar product of the normal vectors. Compare the answers. What does this say about the normal vectors of planes that intersect in a line?
a) \displaystyle 2 x+y+z=7
\displaystyle 4 x+3 y-3 z=13
\displaystyle 4 x+2 y+2 z=14
b) \displaystyle x+3 y-z=4
\displaystyle 3 x+8 y-4 z=4
\displaystyle x+2 y-2 z=-4
c) \displaystyle x+9 y+3 z=23
\displaystyle x+15 y+3 z=29
\displaystyle 4 x-13 y+12 z=43
d) \displaystyle x-6 y+z=-1
\displaystyle x-y=5
\displaystyle 2 x-12 y+2 z=-2
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&3x+2y-z=-2\\
&2x+y-2z= 7\\
&2x-3y + 4z= -3
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&3x-4y+2z= 1\\
&6x-8y+4z= 10\\
&15x-20y+10z=-3
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&x + 4y + 8z= 1\\
&2y - 4z = 10\\
&0=12
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&x-y+z=20\\
&x + y + 3z = -4\\
&2x -5y-z= -6
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&4x+8y+ 4z= 7\\
&5x+ 10y + 5z = -10\\
&3x-y -4z= 6
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&2x+ 2y + z = 10\\
&5x + 4y-4z = 13\\
&3x+5y-2z= 6
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&2x+5y =1\\
&4y+ z= 1\\
&7x-4z = 1
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&3x-2y-z=0\\
&x -z= 0\\
&3x+2y-5z=0
\end{array}
Determine if the system of planes is consistent or inconsistent. If possible, solve the system.
\displaystyle
\begin{array}{llllllll}
&y + 2z= 11\\
&4y -4z = -16\\
&3y-3z= -12
\end{array}
A student solved three systems of planes and obtained these algebraic solutions. Interpret how the planes intersect as exactly as you can in each case.
\displaystyle
\begin{array}{llllllll}
&x = 5\\
&y = 10\\
&z = -3
\end{array}
A student solved three systems of planes and obtained these algebraic solutions. Interpret how the planes intersect as exactly as you can in each case.
\displaystyle
\begin{array}{llllllll}
&x -y+2z=4
&y-3z=8
&0=0
\end{array}
In each case, describe all the ways in which three planes could intersect.
a) The normals are not parallel.
b) Two of three three normals are parallel.
c) All three of the normals are parallel.
Show that the planes in each set are mutually perpendicular and have a unique solution.
\displaystyle
\begin{array}{llllllll}
&5x + y+4z= 18\\
&x+3y-2z= -16\\
&x -y-z=0
\end{array}
Show that the planes in the set are mutually perpendicular and have a unique solution.
\displaystyle
\begin{array}{llllllll}
&2x-6y+z= -16\\
&7x+3y+4z = 41\\
&27x + y - 48 z = -11
\end{array}
You are given the following two planes:
\displaystyle
\begin{array}{llllllll}
&x+4y-3z-12= 0\\
&x + 6y-2z-22= 0
\end{array}
a) Determine if the planes are parallel.
b) Find the line of intersection of the two planes.
c) Use 3-D graphing technology to check your answer to part a).
d) Use the two original equations to determine two other equations that have the same solution as the original two.
e) Verify your answers in part d) by graphing.
f) Find a third equation that will have a unique solution with the original two equations.
A dependent system of equations is one whose solution requires a parameter to express it. Change one of the coefficients in the following system of planes so that the solution is consistent and dependent.
a)
\displaystyle x-3 z=-3,\displaystyle y+5 z=20,\displaystyle 3 x+5 z=3b)
\displaystyle 2 x+8 y=-6,\displaystyle 2 x-z=4,\displaystyle 3 x+12 y+6 z=-9
Determine the equations of three planes that
are all parallel but distinct.
Determine the equations of three planes that
intersect at the point P(3, 1, -9).
Determine the equations of three planes that
intersect in a sheaf of planes
Determine the equations of three planes that
intersect in pairs
Determine the equations of three planes that
intersect in the line (x, y, z)= (1, ,3, -4)+ t(4, 1, 9).
Determine the equations of three planes that
intersect in pairs and are all parallel to the y-axis
Determine the equations of three planes that
intersect at the point (-2, 4, -4) and are all perpendicular to each other
Determine the equations of three planes that
intersect along the z-axis.
Find the volume of the figure bounded by the following planes.
\displaystyle
\begin{array}{llllllll}
&x + z = -3\\
&10x -3z = 22\\
&4x - 9z =-38\\
&y = -4\\
&y = 10
\end{array}
Solve the following system of equations.
\displaystyle
\begin{array}{llllllll}
&2w + x+y+2z = - 9\\
&w-x + y + 2z= 1\\
&2w +x+ 2y - 3z = 18\\
&3w+2x + 3y + z = 0
\end{array}
Create a four—dimensional system that has a solution (3, 1, -4, 6).
A canoeist is crossing a river that is 77 m wide. She is paddling at 7 m/s. The current is 7 m/s (downstream). The canoeist heads out at a 77° angle (upstream). How far down the opposite shore will the canoeist be when she gets to the other side of the river?
The parallelepiped formed by the vectors \vec{a} = [1,2,-3], \vec{b} = [1, k , -3], and \vec{c} = [2, k, 1] has volume 147 unit^3. Determine the possible values of k.