تكلم مع الذكاء الاصطناعي
Deck 13: Vector Geometry
ملء الشاشة (f)
سؤال
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 
سؤال
Let 
. Calculate the following. 
. Calculate the following. سؤال
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 . سؤال
Find 
for which 
is parallel to 
where 
for which is parallel to where سؤال
Express 
as a linear combination of the two vectors 
and 
.
as a linear combination of the two vectors and . سؤال
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)
سؤال
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. سؤال
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 سؤال
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 . سؤال
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).
سؤال
Express 
as a linear combination of the two vectors 
and 
.
as a linear combination of the two vectors and . سؤال
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
سؤال
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 . سؤال
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 سؤال
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 . سؤال
Let 
. Find the vector 
such that 
.
. Find the vector such that . سؤال
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 . سؤال
Express 
as a linear combination of 
and 
.
as a linear combination of and . سؤال
Find 
and 
such that 
is a unit vector parallel to 
where 
and 
.
and such that is a unit vector parallel to where and . سؤال
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? سؤال
Find the decomposition 
of 
with respect to 
.
of with respect to . سؤال
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 سؤال
Find the decomposition 
of 
with respect to 
.
of with respect to . سؤال
Find an equation of the line that passes through the points 
and 
.
and . سؤال
Find a unit vector 
orthogonal to 
and making an angle of 
with the vector 
.
orthogonal to and making an angle of with the vector . سؤال
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 . سؤال
Find the decomposition 
of 
with respect to 
.
of with respect to . سؤال
Find an equation of the line that passes through the points 
and 
.
and . سؤال
Find a scalar 
such that the vectors 
and 
are orthogonal.
such that the vectors and are orthogonal. سؤال
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 . سؤال
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)
سؤال
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 سؤال
Find the angle between the vectors 
and 
.
and . سؤال
Find an equation of the bisector 
of the angle 
in the triangle with vertices 
of the angle in the triangle with vertices سؤال
The angle between the vectors 
and 
is:
A)
.
B)
.
C)
.
D)
.
E)
.
and is:A)
.B)
.C)
.D)
.E)
. سؤال
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 سؤال
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
سؤال
Determine whether the lines 
and 
intersect, and if so, find the point of intersection.
and intersect, and if so, find the point of intersection. سؤال
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 . سؤال
Find an equation of the line that passes through 
and is orthogonal to the vectors 
and 
.
and is orthogonal to the vectors and . سؤال
What is the minimum force you must apply to pull a 35-kg mass up a frictionless ramp, inclined at an angle 
?
? سؤال
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 ? سؤال
Compute the height 
to the side 
in the triangle with vertices at 
.
to the side in the triangle with vertices at . سؤال
Compute the area of the projection of the parallelogram with vertices 
, 
, 
, and 
on the xy-plane.
, , , and on the xy-plane. سؤال
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.
سؤال
Find all the vectors 
of length 
that are orthogonal to the vectors 
and 
.
of length that are orthogonal to the vectors and . سؤال
Determine whether the vectors 
, and 
lie in one plane.
, and lie in one plane. سؤال
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 . سؤال
Find the volume of the parallelepiped with vertices at 
, and 
.
, and . سؤال
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 . سؤال
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 . سؤال
Calculate the volume of the parallelepiped spanned by 
, 
, and 
.
, , and . سؤال
Compute the area of the projection onto the xy-plane of the parallelogram spanned by the vectors 
and 
.
and . سؤال
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)
سؤال
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. سؤال
Compute the height 
to the side 
in the triangle with vertices at 
, 
, and 
.
to the side in the triangle with vertices at , , and . سؤال
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 . سؤال
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 . سؤال
Find the projection of 
along 
if 
and 
.
along if and . سؤال
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 . سؤال
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. سؤال
Describe the trace obtained by intersecting the quadric surface 
with the plane 
with the plane سؤال
Find an equation of the plane with traces 
, and 
in the 
plane, 
plane, and 
plane, respectively.
, and in the plane, plane, and plane, respectively. سؤال
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? سؤال
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 . سؤال
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. سؤال
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 . سؤال
Write the equation of the quadric surface 
in standard form, and identify its type.
in standard form, and identify its type. سؤال
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
سؤال
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.
سؤال
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. سؤال
Find an equation of the plane with traces 
, and 
in the 
plane, 
plane, and 
plane, respectively.
, and in the plane, plane, and plane, respectively. سؤال
For which values of h is the intersection of the horizontal plane 
and the ellipsoid 
empty?
and the ellipsoid empty? سؤال
Write the equation of the quadric surface 
in standard form, and identify its type.
in standard form, and identify its type. سؤال
Identify the quadric surface 
.
. سؤال
Describe the trace obtained by intersecting the quadric surface 
with the plane 
.
with the plane . سؤال
Find an equation of the plane that passes through the points 
, and 
.
, and . سؤال
Find an equation of the plane that passes through the points 
, and 
.
, and . سؤال
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.
سؤال
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.
فتح الحزمة
قم بالتسجيل لفتح البطاقات في هذه المجموعة!
Unlock Deck
Unlock Deck
1/96
العب
ملء الشاشة (f)
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 فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
5
Express as a linear combination of the two vectors and .
as a linear combination of the two vectors and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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)
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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).
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
11
Express as a linear combination of the two vectors and .
as a linear combination of the two vectors and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
16
Let . Find the vector such that .
. Find the vector such that . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
18
Express as a linear combination of and .
as a linear combination of and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
19
Find and such that is a unit vector parallel to where and .
and such that is a unit vector parallel to where and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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? فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
21
Find the decomposition of with respect to .
of with respect to . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
23
Find the decomposition of with respect to .
of with respect to . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
24
Find an equation of the line that passes through the points and .
and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
27
Find the decomposition of with respect to .
of with respect to . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
28
Find an equation of the line that passes through the points and .
and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
29
Find a scalar such that the vectors and are orthogonal.
such that the vectors and are orthogonal. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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)
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
33
Find the angle between the vectors and .
and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
34
Find an equation of the bisector of the angle in the triangle with vertices
of the angle in the triangle with vertices فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
35
The angle between the vectors and is:
A) .
B) .
C) .
D) .
E) .
and is:A)
.B)
.C)
.D)
.E)
. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
40
Find an equation of the line that passes through and is orthogonal to the vectors and .
and is orthogonal to the vectors and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
41
What is the minimum force you must apply to pull a 35-kg mass up a frictionless ramp, inclined at an angle ?
? فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 ? فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
43
Compute the height to the side in the triangle with vertices at .
to the side in the triangle with vertices at . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
44
Compute the area of the projection of the parallelogram with vertices , , , and on the xy-plane.
, , , and on the xy-plane. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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.
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
46
Find all the vectors of length that are orthogonal to the vectors and .
of length that are orthogonal to the vectors and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
47
Determine whether the vectors , and lie in one plane.
, and lie in one plane. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
49
Find the volume of the parallelepiped with vertices at , and .
, and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
52
Calculate the volume of the parallelepiped spanned by , , and .
, , and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
53
Compute the area of the projection onto the xy-plane of the parallelogram spanned by the vectors and .
and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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)
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
56
Compute the height to the side in the triangle with vertices at , , and .
to the side in the triangle with vertices at , , and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
59
Find the projection of along if and .
along if and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
62
Describe the trace obtained by intersecting the quadric surface with the plane
with the plane فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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? فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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 . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
68
Write the equation of the quadric surface in standard form, and identify its type.
in standard form, and identify its type. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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.
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
73
For which values of h is the intersection of the horizontal plane and the ellipsoid empty?
and the ellipsoid empty? فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
74
Write the equation of the quadric surface in standard form, and identify its type.
in standard form, and identify its type. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
75
Identify the quadric surface .
. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
76
Describe the trace obtained by intersecting the quadric surface with the plane .
with the plane . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
77
Find an equation of the plane that passes through the points , and .
, and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
78
Find an equation of the plane that passes through the points , and .
, and . فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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.
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
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. فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
فتح الحزمة
k this deck
فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 96 في هذه المجموعة.
