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Deck 13: Vector Geometry
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Question
In the triangle 
, 
is the midpoint of 
and 
is a point on 
such that 
. 
is the intersection of 
and 
. 
A) Express
in terms of 
, and 
B) Express
in terms of 
, and 
C) Write an equation in terms of
, and 
and use the fact that 
and 
are nonzero, distinct, and nonparallel to find 
and 
, is the midpoint of and is a point on such that . is the intersection of and . A) Express
in terms of , and B) Express
in terms of , and C) Write an equation in terms of
, and and use the fact that and are nonzero, distinct, and nonparallel to find and 
Question
Let 
. Calculate the following. 
. Calculate the following. Question
Find the point of intersection 
of the three medians in the triangle with vertices 
, and 
.
of the three medians in the triangle with vertices , and . Question
Find 
for which 
is parallel to 
where 
for which is parallel to where Question
Express 
as a linear combination of the two vectors 
and 
.
as a linear combination of the two vectors and . Question
On a two-dimensional Cartesian coordinate system, 
is the position vector from the origin to the point 
and 
is the position vector with length 
at an angle 
to the positive x-axis. Find the vectors.
A)
B)
is the position vector from the origin to the point and is the position vector with length at an angle to the positive x-axis. Find the vectors. A)
B)
Question
Find scalars 
, and 
such that:
A)
is parallel to 
.
B)
has the same length as 
.
C)
is a unit vector.
, and such that: A)
is parallel to . B)
has the same length as . C)
is a unit vector. Question
Consider the pentagon below. 
A) Find
in terms of 
, and 
B) Find
in terms of 
, and 
C) Find
in terms of 
and 
A) Find
in terms of , and B) Find
in terms of , and C) Find
in terms of and Question
In the parallelogram 
, shown in the figure below, the following holds. 
A) Write
in terms of 
and 
.
B) Write
in terms of 
and 
.
, shown in the figure below, the following holds. A) Write
in terms of and . B) Write
in terms of and . Question
is a triangle and 
are the midpoints of 
, and 
respectively. 
A) Find the vectors
, and 
in terms of 
and 
B) Compute the sum of the vectors in (A).
Question
Express 
as a linear combination of the two vectors 
and 
.
as a linear combination of the two vectors and . Question
In the triangle with vertices 
, and 
, 
is a point on the side 
, such that 
. 
is a line through 
, parallel to the side 
, intersecting 
at a point 
.
A) Find the vector equation of the line
B) Compute the length of the vector
, and , is a point on the side , such that . is a line through , parallel to the side , intersecting at a point . A) Find the vector equation of the line
B) Compute the length of the vector
Question
Let 
be the line 
.
A) Find the distance
between any point on 
and the origin as a function of 
B) Find the point on
closest to the origin by minimizing 
.
be the line . A) Find the distance
between any point on and the origin as a function of B) Find the point on
closest to the origin by minimizing . Question
Consider the following two lines. 
The two lines intersect if:
A)
.
B)
.
C)
.
D)
.
E)
and 
do not intersect for any value of 
The two lines intersect if:A)
.B)
.C)
.D)
.E)
and do not intersect for any value of Question
Two different nonzero vectors, 
and 
that are not parallel satisfy the equality 
, where 
and 
are scalars.
Find
and 
.
and that are not parallel satisfy the equality , where and are scalars.Find
and . Question
Let 
. Find the vector 
such that 
.
. Find the vector such that . Question
Let 
and 
.
A) Write the coordinates of the point
on 
lying three-fifths of the way from 
to 
.
B) Write
as a linear combination of 
and 
.
and . A) Write the coordinates of the point
on lying three-fifths of the way from to . B) Write
as a linear combination of and . Question
Express 
as a linear combination of 
and 
.
as a linear combination of and . Question
Find 
and 
such that 
is a unit vector parallel to 
where 
and 
.
and such that is a unit vector parallel to where and . Question
Consider the four points 
, 
, 
, and 
. Are 
and 
parallel, and if so, do they point in the same direction?
, , , and . Are and parallel, and if so, do they point in the same direction? Question
Find the decomposition 
of 
with respect to 
.
of with respect to . Question
The two lines 
and 
intersect if:
A)
.
B)
.
C)
.
D)
.
E)
and 
do not intersect for any value of 
and intersect if:A)
.B)
.C)
.D)
.E)
and do not intersect for any value of Question
Find the decomposition 
of 
with respect to 
.
of with respect to . Question
Find an equation of the line that passes through the points 
and 
.
and . Question
Find a unit vector 
orthogonal to 
and making an angle of 
with the vector 
.
orthogonal to and making an angle of with the vector . Question
Let 
and 
. Find the point on 
lying three-fifths of the way from 
to 
.
and . Find the point on lying three-fifths of the way from to . Question
Find the decomposition 
of 
with respect to 
.
of with respect to . Question
Find an equation of the line that passes through the points 
and 
.
and . Question
Find a scalar 
such that the vectors 
and 
are orthogonal.
such that the vectors and are orthogonal. Question
Consider the points 
, 
, 
, and 
. Find the length of the vector starting at the midpoint of 
and ending at the midpoint of 
.
, , , and . Find the length of the vector starting at the midpoint of and ending at the midpoint of . Question
The decomposition 
of 
with respect to 
is which of the following?
A)
B)
C)
D)
E)
of with respect to is which of the following?A)
B)
C)
D)
E)
Question
The angle between 
and 
is 
and 
Let 
and 
be the vectors 
and 
.
A) Find the dot product
B) Find the dot product
and use it to compute the lengths 
and 
C) Find the angle between
and 
and is and Let and be the vectors and . A) Find the dot product
B) Find the dot product
and use it to compute the lengths and C) Find the angle between
and Question
Find the angle between the vectors 
and 
.
and . Question
Find an equation of the bisector 
of the angle 
in the triangle with vertices 
of the angle in the triangle with vertices Question
The angle between the vectors 
and 
is:
A)
.
B)
.
C)
.
D)
.
E)
.
and is:A)
.B)
.C)
.D)
.E)
. Question
The two lines 
and 
intersect if:
A)
.
B)
.
C)
.
D)
.
E)
and 
do not intersect for any value of 
and intersect if:A)
.B)
.C)
.D)
.E)
and do not intersect for any value of Question
In the triangle with vertices 
, and 
, 
is a point on 
such that 
. 
is a line through 
, parallel to 
and intersecting 
at point 
.
A) Find a vector equation of the line
B) Find the length of
, and , is a point on such that . is a line through , parallel to and intersecting at point . A) Find a vector equation of the line
B) Find the length of
Question
Determine whether the lines 
and 
intersect, and if so, find the point of intersection.
and intersect, and if so, find the point of intersection. Question
The angle between two intersecting lines is the acute angle between the direction vectors of the lines. Find the angle between the line through the points 
and 
and the line through the points 
and 
.
and and the line through the points and . Question
Find an equation of the line that passes through 
and is orthogonal to the vectors 
and 
.
and is orthogonal to the vectors and . Question
What is the minimum force you must apply to pull a 35-kg mass up a frictionless ramp, inclined at an angle 
?
? Question
Consider the parallel lines 
and 
Compute the distance 
between the two lines using the following steps.
A) Choose any two points
and 
on 
B) Choose any point
on 
C) Compute the area of the triangle
, the length of 
, and find the height from 
to 
, using the area of a triangle.
D) What is the distance between
and 
?
and Compute the distance between the two lines using the following steps. A) Choose any two points
and on B) Choose any point
on C) Compute the area of the triangle
, the length of , and find the height from to , using the area of a triangle. D) What is the distance between
and ? Question
Compute the height 
to the side 
in the triangle with vertices at 
.
to the side in the triangle with vertices at . Question
Compute the area of the projection of the parallelogram with vertices 
, 
, 
, and 
on the xy-plane.
, , , and on the xy-plane. Question
The height 
of the triangle 
shown in the figure is which of the following? 
A)
B)
C)
D)
E) none of the above.
of the triangle shown in the figure is which of the following? A)
B)
C)
D)
E) none of the above.
Question
Find all the vectors 
of length 
that are orthogonal to the vectors 
and 
.
of length that are orthogonal to the vectors and . Question
Determine whether the vectors 
, and 
lie in one plane.
, and lie in one plane. Question
Let 
A) Find the area of the parallelogram spanned by
and 
.
B) Find a vector
orthogonal to 
and 
so that the volume of the parallelepiped spanned by 
, 
, and 
is 
.
A) Find the area of the parallelogram spanned by
and . B) Find a vector
orthogonal to and so that the volume of the parallelepiped spanned by , , and is . Question
Find the volume of the parallelepiped with vertices at 
, and 
.
, and . Question
The angle between two intersecting lines is the acute angle between the direction vectors of the lines. Find the angle between the line through the points 
and 
and the line through the points 
and 
.
and and the line through the points and . Question
The line 
is determined by the two points 
and 
, and the line 
is determined by the points 
and 
. Find the unit vectors 
along the line perpendicular to both 
and 
.
is determined by the two points and , and the line is determined by the points and . Find the unit vectors along the line perpendicular to both and . Question
Calculate the volume of the parallelepiped spanned by 
, 
, and 
.
, , and . Question
Compute the area of the projection onto the xy-plane of the parallelogram spanned by the vectors 
and 
.
and . Question
Which of the following lines 
is parallel to the plane 
A)
B)
C)
D)
E)
is parallel to the plane A)
B)
C)
D)
E)
Question
Let 
and 
be two nonzero orthogonal vectors.
A) Find
in terms of 
, 
, and 
B) Find the scalar
such that 
C) If
and 
are orthogonal unit vectors, write the area of the parallelogram spanned by 
and 
in terms of 
only.
and be two nonzero orthogonal vectors. A) Find
in terms of , , and B) Find the scalar
such that C) If
and are orthogonal unit vectors, write the area of the parallelogram spanned by and in terms of only. Question
Compute the height 
to the side 
in the triangle with vertices at 
, 
, and 
.
to the side in the triangle with vertices at , , and . Question
The line 
is determined by the points 
and 
.
The line
is determined by the points 
and 
.
Find a unit vector along the line perpendicular to both
and 
.
is determined by the points and .The line
is determined by the points and .Find a unit vector along the line perpendicular to both
and . Question
The line 
is determined by the two points 
and 
and the line 
is determined by the points 
and 
. Find a unit vector along the line perpendicular to both 
and 
.
is determined by the two points and and the line is determined by the points and . Find a unit vector along the line perpendicular to both and . Question
Find the projection of 
along 
if 
and 
.
along if and . Question
For the plane 
and the line 
: 
, find the scalar form of the equation of the plane 
that contains 
and is orthogonal to 
.
and the line : , find the scalar form of the equation of the plane that contains and is orthogonal to . Question
Find an equation of the plane passing through the points 
and 
and with the vector 
lying on the plane.
and and with the vector lying on the plane. Question
Describe the trace obtained by intersecting the quadric surface 
with the plane 
with the plane Question
Find an equation of the plane with traces 
, and 
in the 
plane, 
plane, and 
plane, respectively.
, and in the plane, plane, and plane, respectively. Question
For which values of 
is the intersection of the horizontal plane 
and the hyperboloid 
empty?
is the intersection of the horizontal plane and the hyperboloid empty? Question
Find an equation of the plane passing through the points 
and 
, and parallel to any line with the direction vector 
.
and , and parallel to any line with the direction vector . Question
Write the equation of the quadric surface 
in standard form, and identify the type of the quadric conic.
in standard form, and identify the type of the quadric conic. Question
Let 
be the line 
and 
be the plane 
.
A) Find the point of intersection
of 
and 
.
B) Find an equation of the plane
(in scalar form) that passes through the point 
and is orthogonal to 
.
be the line and be the plane . A) Find the point of intersection
of and . B) Find an equation of the plane
(in scalar form) that passes through the point and is orthogonal to . Question
Write the equation of the quadric surface 
in standard form, and identify its type.
in standard form, and identify its type. Question
The following lines ( 
is a parameter) are known to be in one plane. 
If they intersect at the point 
, find the following.
A) the value of
B) the scalar equation of the plane
is a parameter) are known to be in one plane. If they intersect at the point , find the following. A) the value of
B) the scalar equation of the plane
Question
Consider the 3 lines. 
Which of the following statements is correct?
A) The three lines are parallel.
B) The three lines lie in one plane and do not intersect at one point.
C) The three lines are not contained in one plane.
D) The three lines lie in one plane and they intersect at one point.
E) Two of the lines are parallel.
Which of the following statements is correct?A) The three lines are parallel.
B) The three lines lie in one plane and do not intersect at one point.
C) The three lines are not contained in one plane.
D) The three lines lie in one plane and they intersect at one point.
E) Two of the lines are parallel.
Question
Let 
and 
be the plane and line determined by the following equations 
Find the equation of the line 
passing through the intersection point of 
and 
and parallel to the trace of 
in the 
plane.
and be the plane and line determined by the following equations Find the equation of the line passing through the intersection point of and and parallel to the trace of in the plane. Question
Find an equation of the plane with traces 
, and 
in the 
plane, 
plane, and 
plane, respectively.
, and in the plane, plane, and plane, respectively. Question
For which values of h is the intersection of the horizontal plane 
and the ellipsoid 
empty?
and the ellipsoid empty? Question
Write the equation of the quadric surface 
in standard form, and identify its type.
in standard form, and identify its type. Question
Identify the quadric surface 
.
. Question
Describe the trace obtained by intersecting the quadric surface 
with the plane 
.
with the plane . Question
Find an equation of the plane that passes through the points 
, and 
.
, and . Question
Find an equation of the plane that passes through the points 
, and 
.
, and . Question
Three nonzero vectors, 
in 3-space are in one plane (coplanar) if:
A)
B) one of them is a linear combination of the other two.
C) one of them is parallel to the cross product of the other two.
D) the scalar triple product of
, and 
is zero.
E) both B and D.
in 3-space are in one plane (coplanar) if:A)
B) one of them is a linear combination of the other two.
C) one of them is parallel to the cross product of the other two.
D) the scalar triple product of
, and is zero.E) both B and D.
Question
The plane 
is determined by the points 
, and 
. 
is the trace of 
in the xy-plane. 
is the line 
.
Which of the following statements is correct?
A)
and 
intersect.
B)
and 
are parallel.
C)
and 
do not lie in one plane.
D) There is not enough data to conclude about the mutual position of
and 
.
E)
and 
coincide.
is determined by the points , and . is the trace of in the xy-plane. is the line .Which of the following statements is correct?
A)
and intersect.B)
and are parallel.C)
and do not lie in one plane.D) There is not enough data to conclude about the mutual position of
and .E)
and coincide.
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Deck 13: Vector Geometry
1
In the triangle , is the midpoint of and is a point on such that . is the intersection of and .
A) Express in terms of , and
B) Express in terms of , and
C) Write an equation in terms of , and and use the fact that and are nonzero, distinct, and nonparallel to find and
, is the midpoint of and is a point on such that . is the intersection of and . A) Express
in terms of , and B) Express
in terms of , and C) Write an equation in terms of
, and and use the fact that and are nonzero, distinct, and nonparallel to find and A) 
B) 
C) 
B) C) 2
Let . Calculate the following.
. Calculate the following. A) 
B) 
C) 
B) C) 3
Find the point of intersection of the three medians in the triangle with vertices , and .
of the three medians in the triangle with vertices , and .
4
Find for which is parallel to where
for which is parallel to where Unlock Deck
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5
Express as a linear combination of the two vectors and .
as a linear combination of the two vectors and . Unlock Deck
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6
On a two-dimensional Cartesian coordinate system, is the position vector from the origin to the point and is the position vector with length at an angle to the positive x-axis. Find the vectors.
A)
B)
is the position vector from the origin to the point and is the position vector with length at an angle to the positive x-axis. Find the vectors. A)
B)
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7
Find scalars , and such that:
A) is parallel to .
B) has the same length as .
C) is a unit vector.
, and such that: A)
is parallel to . B)
has the same length as . C)
is a unit vector. Unlock Deck
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8
Consider the pentagon below.
A) Find in terms of , and
B) Find in terms of , and
C) Find in terms of and
A) Find
in terms of , and B) Find
in terms of , and C) Find
in terms of and Unlock Deck
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9
In the parallelogram , shown in the figure below, the following holds.
A) Write in terms of and .
B) Write in terms of and .
, shown in the figure below, the following holds. A) Write
in terms of and . B) Write
in terms of and . Unlock Deck
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10
is a triangle and are the midpoints of , and respectively. A) Find the vectors
, and in terms of and B) Compute the sum of the vectors in (A).
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11
Express as a linear combination of the two vectors and .
as a linear combination of the two vectors and . Unlock Deck
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12
In the triangle with vertices , and , is a point on the side , such that . is a line through , parallel to the side , intersecting at a point .
A) Find the vector equation of the line
B) Compute the length of the vector
, and , is a point on the side , such that . is a line through , parallel to the side , intersecting at a point . A) Find the vector equation of the line
B) Compute the length of the vector
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13
Let be the line .
A) Find the distance between any point on and the origin as a function of
B) Find the point on closest to the origin by minimizing .
be the line . A) Find the distance
between any point on and the origin as a function of B) Find the point on
closest to the origin by minimizing . Unlock Deck
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14
Consider the following two lines. The two lines intersect if:
A) .
B) .
C) .
D) .
E) and do not intersect for any value of
The two lines intersect if:A)
.B)
.C)
.D)
.E)
and do not intersect for any value of Unlock Deck
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15
Two different nonzero vectors, and that are not parallel satisfy the equality , where and are scalars.
Find and .
and that are not parallel satisfy the equality , where and are scalars.Find
and . Unlock Deck
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16
Let . Find the vector such that .
. Find the vector such that . Unlock Deck
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17
Let and .
A) Write the coordinates of the point on lying three-fifths of the way from to .
B) Write as a linear combination of and .
and . A) Write the coordinates of the point
on lying three-fifths of the way from to . B) Write
as a linear combination of and . Unlock Deck
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18
Express as a linear combination of and .
as a linear combination of and . Unlock Deck
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19
Find and such that is a unit vector parallel to where and .
and such that is a unit vector parallel to where and . Unlock Deck
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20
Consider the four points , , , and . Are and parallel, and if so, do they point in the same direction?
, , , and . Are and parallel, and if so, do they point in the same direction? Unlock Deck
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21
Find the decomposition of with respect to .
of with respect to . Unlock Deck
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22
The two lines and intersect if:
A) .
B) .
C) .
D) .
E) and do not intersect for any value of
and intersect if:A)
.B)
.C)
.D)
.E)
and do not intersect for any value of Unlock Deck
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23
Find the decomposition of with respect to .
of with respect to . Unlock Deck
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24
Find an equation of the line that passes through the points and .
and . Unlock Deck
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25
Find a unit vector orthogonal to and making an angle of with the vector .
orthogonal to and making an angle of with the vector . Unlock Deck
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26
Let and . Find the point on lying three-fifths of the way from to .
and . Find the point on lying three-fifths of the way from to . Unlock Deck
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27
Find the decomposition of with respect to .
of with respect to . Unlock Deck
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28
Find an equation of the line that passes through the points and .
and . Unlock Deck
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29
Find a scalar such that the vectors and are orthogonal.
such that the vectors and are orthogonal. Unlock Deck
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30
Consider the points , , , and . Find the length of the vector starting at the midpoint of and ending at the midpoint of .
, , , and . Find the length of the vector starting at the midpoint of and ending at the midpoint of . Unlock Deck
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31
The decomposition of with respect to is which of the following?
A)
B)
C)
D)
E)
of with respect to is which of the following?A)
B)
C)
D)
E)
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32
The angle between and is and Let and be the vectors and .
A) Find the dot product
B) Find the dot product and use it to compute the lengths and
C) Find the angle between and
and is and Let and be the vectors and . A) Find the dot product
B) Find the dot product
and use it to compute the lengths and C) Find the angle between
and Unlock Deck
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33
Find the angle between the vectors and .
and . Unlock Deck
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34
Find an equation of the bisector of the angle in the triangle with vertices
of the angle in the triangle with vertices Unlock Deck
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35
The angle between the vectors and is:
A) .
B) .
C) .
D) .
E) .
and is:A)
.B)
.C)
.D)
.E)
. Unlock Deck
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36
The two lines and intersect if:
A) .
B) .
C) .
D) .
E) and do not intersect for any value of
and intersect if:A)
.B)
.C)
.D)
.E)
and do not intersect for any value of Unlock Deck
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37
In the triangle with vertices , and , is a point on such that . is a line through , parallel to and intersecting at point .
A) Find a vector equation of the line
B) Find the length of
, and , is a point on such that . is a line through , parallel to and intersecting at point . A) Find a vector equation of the line
B) Find the length of
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38
Determine whether the lines and intersect, and if so, find the point of intersection.
and intersect, and if so, find the point of intersection. Unlock Deck
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39
The angle between two intersecting lines is the acute angle between the direction vectors of the lines. Find the angle between the line through the points and and the line through the points and .
and and the line through the points and . Unlock Deck
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40
Find an equation of the line that passes through and is orthogonal to the vectors and .
and is orthogonal to the vectors and . Unlock Deck
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41
What is the minimum force you must apply to pull a 35-kg mass up a frictionless ramp, inclined at an angle ?
? Unlock Deck
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42
Consider the parallel lines and Compute the distance between the two lines using the following steps.
A) Choose any two points and on
B) Choose any point on
C) Compute the area of the triangle , the length of , and find the height from to , using the area of a triangle.
D) What is the distance between and ?
and Compute the distance between the two lines using the following steps. A) Choose any two points
and on B) Choose any point
on C) Compute the area of the triangle
, the length of , and find the height from to , using the area of a triangle. D) What is the distance between
and ? Unlock Deck
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43
Compute the height to the side in the triangle with vertices at .
to the side in the triangle with vertices at . Unlock Deck
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44
Compute the area of the projection of the parallelogram with vertices , , , and on the xy-plane.
, , , and on the xy-plane. Unlock Deck
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45
The height of the triangle shown in the figure is which of the following?
A)
B)
C)
D)
E) none of the above.
of the triangle shown in the figure is which of the following? A)
B)
C)
D)
E) none of the above.
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46
Find all the vectors of length that are orthogonal to the vectors and .
of length that are orthogonal to the vectors and . Unlock Deck
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47
Determine whether the vectors , and lie in one plane.
, and lie in one plane. Unlock Deck
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48
Let
A) Find the area of the parallelogram spanned by and .
B) Find a vector orthogonal to and so that the volume of the parallelepiped spanned by , , and is .
A) Find the area of the parallelogram spanned by
and . B) Find a vector
orthogonal to and so that the volume of the parallelepiped spanned by , , and is . Unlock Deck
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49
Find the volume of the parallelepiped with vertices at , and .
, and . Unlock Deck
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50
The angle between two intersecting lines is the acute angle between the direction vectors of the lines. Find the angle between the line through the points and and the line through the points and .
and and the line through the points and . Unlock Deck
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51
The line is determined by the two points and , and the line is determined by the points and . Find the unit vectors along the line perpendicular to both and .
is determined by the two points and , and the line is determined by the points and . Find the unit vectors along the line perpendicular to both and . Unlock Deck
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52
Calculate the volume of the parallelepiped spanned by , , and .
, , and . Unlock Deck
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53
Compute the area of the projection onto the xy-plane of the parallelogram spanned by the vectors and .
and . Unlock Deck
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54
Which of the following lines is parallel to the plane
A)
B)
C)
D)
E)
is parallel to the plane A)
B)
C)
D)
E)
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55
Let and be two nonzero orthogonal vectors.
A) Find in terms of , , and
B) Find the scalar such that
C) If and are orthogonal unit vectors, write the area of the parallelogram spanned by and in terms of only.
and be two nonzero orthogonal vectors. A) Find
in terms of , , and B) Find the scalar
such that C) If
and are orthogonal unit vectors, write the area of the parallelogram spanned by and in terms of only. Unlock Deck
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56
Compute the height to the side in the triangle with vertices at , , and .
to the side in the triangle with vertices at , , and . Unlock Deck
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57
The line is determined by the points and .
The line is determined by the points and .
Find a unit vector along the line perpendicular to both and .
is determined by the points and .The line
is determined by the points and .Find a unit vector along the line perpendicular to both
and . Unlock Deck
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58
The line is determined by the two points and and the line is determined by the points and . Find a unit vector along the line perpendicular to both and .
is determined by the two points and and the line is determined by the points and . Find a unit vector along the line perpendicular to both and . Unlock Deck
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59
Find the projection of along if and .
along if and . Unlock Deck
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60
For the plane and the line : , find the scalar form of the equation of the plane that contains and is orthogonal to .
and the line : , find the scalar form of the equation of the plane that contains and is orthogonal to . Unlock Deck
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61
Find an equation of the plane passing through the points and and with the vector lying on the plane.
and and with the vector lying on the plane. Unlock Deck
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62
Describe the trace obtained by intersecting the quadric surface with the plane
with the plane Unlock Deck
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63
Find an equation of the plane with traces , and in the plane, plane, and plane, respectively.
, and in the plane, plane, and plane, respectively. Unlock Deck
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64
For which values of is the intersection of the horizontal plane and the hyperboloid empty?
is the intersection of the horizontal plane and the hyperboloid empty? Unlock Deck
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65
Find an equation of the plane passing through the points and , and parallel to any line with the direction vector .
and , and parallel to any line with the direction vector . Unlock Deck
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66
Write the equation of the quadric surface in standard form, and identify the type of the quadric conic.
in standard form, and identify the type of the quadric conic. Unlock Deck
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67
Let be the line and be the plane .
A) Find the point of intersection of and .
B) Find an equation of the plane (in scalar form) that passes through the point and is orthogonal to .
be the line and be the plane . A) Find the point of intersection
of and . B) Find an equation of the plane
(in scalar form) that passes through the point and is orthogonal to . Unlock Deck
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68
Write the equation of the quadric surface in standard form, and identify its type.
in standard form, and identify its type. Unlock Deck
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69
The following lines ( is a parameter) are known to be in one plane. If they intersect at the point , find the following.
A) the value of
B) the scalar equation of the plane
is a parameter) are known to be in one plane. If they intersect at the point , find the following. A) the value of
B) the scalar equation of the plane
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70
Consider the 3 lines. Which of the following statements is correct?
A) The three lines are parallel.
B) The three lines lie in one plane and do not intersect at one point.
C) The three lines are not contained in one plane.
D) The three lines lie in one plane and they intersect at one point.
E) Two of the lines are parallel.
Which of the following statements is correct?A) The three lines are parallel.
B) The three lines lie in one plane and do not intersect at one point.
C) The three lines are not contained in one plane.
D) The three lines lie in one plane and they intersect at one point.
E) Two of the lines are parallel.
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71
Let and be the plane and line determined by the following equations Find the equation of the line passing through the intersection point of and and parallel to the trace of in the plane.
and be the plane and line determined by the following equations Find the equation of the line passing through the intersection point of and and parallel to the trace of in the plane. Unlock Deck
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72
Find an equation of the plane with traces , and in the plane, plane, and plane, respectively.
, and in the plane, plane, and plane, respectively. Unlock Deck
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73
For which values of h is the intersection of the horizontal plane and the ellipsoid empty?
and the ellipsoid empty? Unlock Deck
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74
Write the equation of the quadric surface in standard form, and identify its type.
in standard form, and identify its type. Unlock Deck
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75
Identify the quadric surface .
. Unlock Deck
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76
Describe the trace obtained by intersecting the quadric surface with the plane .
with the plane . Unlock Deck
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77
Find an equation of the plane that passes through the points , and .
, and . Unlock Deck
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78
Find an equation of the plane that passes through the points , and .
, and . Unlock Deck
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79
Three nonzero vectors, in 3-space are in one plane (coplanar) if:
A)
B) one of them is a linear combination of the other two.
C) one of them is parallel to the cross product of the other two.
D) the scalar triple product of , and is zero.
E) both B and D.
in 3-space are in one plane (coplanar) if:A)
B) one of them is a linear combination of the other two.
C) one of them is parallel to the cross product of the other two.
D) the scalar triple product of
, and is zero.E) both B and D.
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80
The plane is determined by the points , and . is the trace of in the xy-plane. is the line .
Which of the following statements is correct?
A) and intersect.
B) and are parallel.
C) and do not lie in one plane.
D) There is not enough data to conclude about the mutual position of and .
E) and coincide.
is determined by the points , and . is the trace of in the xy-plane. is the line .Which of the following statements is correct?
A)
and intersect.B)
and are parallel.C)
and do not lie in one plane.D) There is not enough data to conclude about the mutual position of
and .E)
and coincide. Unlock Deck
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Unlock Deck
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