A line in two-space has scalar equation
3 x+5 y+9=0 . Which vector is parallel to the
line?
A [3,5]
B \displaystyle
[3,-5]
C [5,3]
D \displaystyle
[-5,3]
Which line is the same as
\displaystyle
[x, y]=[1,-8]+t[4,-3] ?
\displaystyle
\mathbf{A} [x, y]=[2,4]+t[-4,3]
\displaystyle
\mathbf{B}[x, y]=[13,-17]+t[8,-6]
\displaystyle
\mathbf{C}[x, y]=[-10,20]+t[12,-9]
\displaystyle
\mathbf{D}[x, y]=[-11,-4]+t[4,-3]
Which plane does not contain the point
\displaystyle
\mathrm{P } (10,-3,5) ?
\displaystyle
\mathbf{A } 3 x+6 y-2 z-2=0
\displaystyle
\mathbf{B } x+y-z-12=0
\displaystyle
\mathbf{C } 2 x-2 y-3 z-11=0
\displaystyle
\mathbf{D} 4 x+5 y+z-30=0
Which does not exist for a line in three-space?
\mathbf{A} vector equation
\mathbf{B} scalar equation
\mathbf{C} parametric equation
\mathbf{D} a point and a direction vector
The equation 2 x+5 y=9 can represent
A a scalar equation of a line in two-space
B a scalar equation of a plane in three-space
C neither A nor B
D both A and B
Which vector is not normal to the plane
\displaystyle
x+2 y-3 z-4=0 ?
\displaystyle
\mathbf{A }[2,-3,4]
\displaystyle
\mathbf{B }[-1,-2,3]
\displaystyle
\mathbf{C }[2,4,-6]
\displaystyle
\mathbf{D }[3,6,-9]
Which point is not a solution to the equation
\displaystyle
3 x-4 y+z-12=0 ?
\displaystyle
\mathbf{A}(-8,-9,0)
\displaystyle
\mathbf{B} (4,1,4)
\displaystyle
\mathbf{C} (16,10,4)
\displaystyle
\mathbf{D} (18,12,2)
How many solutions does this system have?
\displaystyle
10 x-7 y=10
\displaystyle
4 x+5 y=12
\mathbf{A} 0
\displaystyle
\mathbf{B} 1
\mathbf{C} 12
\mathbf{D} infinitely many
Which word best describes the solution to this
system of equations?
\displaystyle
[x, y, z]=[4,-2,3]+s[5,-3,2]
\displaystyle
[x, y, z]=[1,-3,1]+t[5,-3,2]
A consistent
B coincident
C inconsistent
D skew
Which statement is never true for two planes?
A they intersect at a point
B they intersect at a line
C they are parallel and distinct
D they are coincident
A line passes through points \mathrm{A}(1,5,-4) and
\displaystyle
\mathrm{B}(2,-9,0)
a) Write vector and parametric equations of the line.
b) Determine two other points on the line.
Write the parametric equations of a line
perpendicular to 4 x+8 y+7=0 with the same
x-intercept as
\displaystyle
[x, y]=[2,7]+t[-10,3] .
Find the parametric equations of the line
through the point \mathrm{P}(-6,4,3) , that is
perpendicular to both of the lines
\displaystyle
[x, y, z]=[0,-10,-2]+s[4,6,-3]
\displaystyle
[x, y, z]=[5,5,-5]+t[3,2,4]
Write the scalar equation of a plane that is
parallel to the x z -plane and contains the line
\displaystyle
[x, y, z]=[3,-1,5]+t[4,0,-1]
Determine if the lines in each pair intersect. If they intersect, find the intersection point.
\displaystyle
6 x+2 y=5
\displaystyle
[x, y]=[4,-7]+t[1,-3]
Determine if the lines in each pair intersect. If they intersect, find the intersection point.
\displaystyle
2 x+3 y=21
\displaystyle
4 x-y=7
Determine if the lines in each pair intersect. If they intersect, find the intersection point.
[x, y, z]=[-2,4,-1]+s[3,-3,1]
[x, y, z]=[7,10,4]+t[1,4,3]
Determine if the lines in each pair intersect. If they intersect, find the intersection point.
\displaystyle
[x, y, z]=[3,4,-6]+s[5,2,-2]
\displaystyle
[x, y, z]=[1,-4,4]+t[10,4,-4]
Determine whether the line with equation
\displaystyle
[x, y, z]=[-3,-6,-11]+k[22,1,-11]
in the plane that contains the points \mathrm{A}(2,5,6) ,
\mathrm{B}(-7,1,4)
,
\mathrm{C}(6,-2,-9)
Determine if the planes in each set intersect. If so, describe how they intersect.
\displaystyle
2 x+5 y-7 z=31
\displaystyle
x-2 y-5 z=-9
Determine if the planes in each set intersect. If so, describe how they intersect.
\displaystyle
x+y-z=11
\displaystyle
2 x+3 y+4 z=0
\displaystyle
2 x-2 y+z=4
Determine if the planes in each set intersect. If so, describe how they intersect.
\displaystyle
2 x+y-3 z=2
\displaystyle
x-4 y+2 z=5
\displaystyle
4x+2 y-6 z=-12
Determine if the planes in each set intersect. If so, describe how they intersect.
\displaystyle
2 x-3 y-4 z=-16
\displaystyle
11 x-3 y+5 z=47
\displaystyle
5 x+y+7 z=45
A plane passes through the points \mathrm{A}(1,13,2) ,
\displaystyle
\mathrm{B}(-2,-6,5)
\displaystyle
\mathrm{C}(-1,-1,-3)
a) Write the vector and parametric equations of the plane.
b) Write the scalar equation of the plane.
c) Determine two other points on the plane.
The plane \pi has vector equation
\displaystyle
\vec{r}=[2,1,3]+s[1,1,1]+t[2,0,2], s, t \in \mathbb{R}
a) Verify that \pi does not pass through the
origin.
b) Find the distance from the origin to \pi .
A plane \pi_{1} has equation 2 x-3 y+5 z=1 . Two
other planes, \pi_{2} and \pi_{3} , intersect the plane, and
each other, as illustrated in the diagram. State
possible equations for \pi_{2} and \pi_{3} and justify
your reasoning.
Given the plane 2 x+3 y+4 z=24 and the lines
\displaystyle
[x, y, z]=[0,0,6]+r[3,2,-3]
\displaystyle
[x, y, z]=[14,-12,8]+s[1,-2,1]
\displaystyle
[x, y, z]=[9,-10,9]+t[1,6,-5]
a) show that each line is contained in the plane
b) find the vertices of the triangle formed by the lines
c) find the perimeter of the triangle formed by the lines
d) find the area of the triangle formed by the lines