تكلم مع الذكاء الاصطناعي
Deck 12: Vectors and the Geometry of Space
ملء الشاشة (f)
سؤال
Find an equation of the set of all points equidistant from the points 
and 
A)
B)
C)
D)
E)
and A)
B)
C)
D)
E)
سؤال
Reduce the equation to one of the standard forms. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find the point at which the line given by the parametric equations below intersects the plane. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find an equation of the plane that passes through the line of intersection of the planes 
and 
is perpendicular to the plane 
and is perpendicular to the plane سؤال
Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is 
times the distance from P to the yz-plane.
A)
B)
C)
D)
E)
times the distance from P to the yz-plane.A)
B)
C)
D)
E)
سؤال
Find parametric equations for the line through 
and 
and سؤال
Find an equation for the surface obtained by rotating the parabola 
about the y-axis.
A)
B)
C)
D)
E)
about the y-axis.A)
B)
C)
D)
E)
سؤال
Find an equation for the surface obtained by rotating the line 
about the x-axis.
about the x-axis. سؤال
Find the distance between the planes. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find an equation of the plane with x-intercept 
, y-intercept 
and z-intercept 
.
A)
B)
C)
D)
E)
, y-intercept and z-intercept .A)
B)
C)
D)
E)
سؤال
Find an equation for the surface consisting of all points that are equidistant from the point 
and the plane 
.
A)
B)
C)
D)
E)
and the plane .A)
B)
C)
D)
E)
سؤال
Identify the planes that are perpendicular.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find, correct to the nearest degree, the angle between the planes. 
A)
B)
C)
D)
E)none of these
A)
B)
C)
D)
E)none of these
سؤال
Find the distance (correct to two decimal places) between the given parallel planes. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find parametric equations for the line through the point 
that is parallel to the plane 
and perpendicular to the line 
A)
B)
C)
D)
E)
that is parallel to the plane and perpendicular to the line A)
B)
C)
D)
E)
سؤال
Find the equation of the line through 
and perpendicular to the plane 
and perpendicular to the plane سؤال
Which of the given lines is parallel to the line 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Classify the surface. 
A)A circular paraboloid with vertex
and axis the z-axis.
B)A hyperboloid of one sheet with center
and axis parallel to the z-axis.
C)A cone with axis parallel to the z-axis and vertex
.
A)A circular paraboloid with vertex
and axis the z-axis.B)A hyperboloid of one sheet with center
and axis parallel to the z-axis.C)A cone with axis parallel to the z-axis and vertex
. سؤال
An ellipsoid is created by rotating the ellipse 
about the x-axis. Find the equation of the ellipsoid.
A)
B)
C)
D)
E)
about the x-axis. Find the equation of the ellipsoid.A)
B)
C)
D)
E)
سؤال
Find the distance between the point 
and the plane 
Round your answer to two decimal place.
A)
B)
C)
D)
E)
and the plane Round your answer to two decimal place.A)
B)
C)
D)
E)
سؤال
A woman walks due west on the deck of a ship at 
mi/h. The ship is moving north at a speed of 
mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.
mi/h. The ship is moving north at a speed of mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth. سؤال
Find the point of intersection. 
سؤال
Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 
سؤال
Find the values of x such that the vectors 
and 
are orthogonal.
A)
B)
C)
D)
E)
and are orthogonal.A)
B)
C)
D)
E)
سؤال
Let 
and let u be a vector with length 
that starts at the origin and rotates in the xy - plane. Find the maximum value of the length of the vector 
.
A)
B)
C)
D)
E)
and let u be a vector with length that starts at the origin and rotates in the xy - plane. Find the maximum value of the length of the vector .A)
B)
C)
D)
E)
سؤال
Find the work done by a force 
that moves an object from the point 
to the point 
along a straight line. The distance is measured in meters and the force in newtons.
A)
B)
C)
D)
E)
that moves an object from the point to the point along a straight line. The distance is measured in meters and the force in newtons.A)
B)
C)
D)
E)
سؤال
Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as shown. Let 
ft, 
ft, 
, 
lb. 
ft, ft, , lb. سؤال
Find 
correct to three decimal places where 
, 
, 
.
A)
B)
C)
D)
E)
correct to three decimal places where , , .A)
B)
C)
D)
E)
سؤال
The tension T at each end of the chain has magnitude 
N and makes an angle 
with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth. 
N and makes an angle with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth. سؤال
Find an equation of the plane through the origin and parallel to the plane 
سؤال
Find the cross product 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Find parametric equations for the line through 
and parallel to the vector 
and parallel to the vector سؤال
A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is 
cm long.
Find the magnitude of the torque about P correct to two decimal places.
Let
N, 
, 
. 
cm long.Find the magnitude of the torque about P correct to two decimal places.
Let
N, , . سؤال
Two forces 
with magnitudes 8 lb and 12 lb act on an object at a point P as shown in the figure. Find the magnitude of the resultant force F acting at P. Round the result to the nearest tenth. 
A)
B)
C)
D)
E)
with magnitudes 8 lb and 12 lb act on an object at a point P as shown in the figure. Find the magnitude of the resultant force F acting at P. Round the result to the nearest tenth. A)
B)
C)
D)
E)
سؤال
Find a nonzero vector orthogonal to the plane through the points P, Q, and R. 
A)
B)
C)
D)
E)None of these
A)
B)
C)
D)
E)None of these
سؤال
Find the unit vectors that are parallel to the tangent line to the curve 
at the point 
.
A)
B)
C)
D)
E)
at the point .A)
B)
C)
D)
E)
سؤال
Velocities have both direction and magnitude and thus are vectors. The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N 
W at a speed of 
km/h. (This means that the direction from which the wind blows is 
west of the northerly direction.) A pilot is steering a plane in the direction N 
E at an airspeed (speed in still air) of 
km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the ground speed of the plane. Round the result to the nearest hundredth.
W at a speed of km/h. (This means that the direction from which the wind blows is west of the northerly direction.) A pilot is steering a plane in the direction N E at an airspeed (speed in still air) of km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the ground speed of the plane. Round the result to the nearest hundredth. سؤال
Calculate the given quantities if 
سؤال
Calculate the angle between a and b (correct to the nearest degree). 
A)
B)
C)
D)
E)None of these
A)
B)
C)
D)
E)None of these
سؤال
Find an equation of the plane that passes through the point 
and contains the line 
and contains the line سؤال
Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. A (-1, 0, 1), B (1, 1, -1), C (1, 1, 1)
A)1, 1,
, both
B)3, 2,
, right
C)1, 2,
, neither
D)1, 1,
, isosceles
A)1, 1,
, bothB)3, 2,
, rightC)1, 2,
, neitherD)1, 1,
, isosceles سؤال
Draw a rectangular box with the origin and 
as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.
as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box. سؤال
Find an equation of the sphere that passes through the point 
and has center 
.
and has center . سؤال
Find the center and the radius of the sphere that has the given equation. 
+ 
+ 
- 6x + 4y = 0
A)( -3, 2, 0),
B)( 3, -2, 0), 13
C)( -3, 2, 0), 13
D)( 3, -2, 0),
+ + - 6x + 4y = 0A)( -3, 2, 0),
B)( 3, -2, 0), 13
C)( -3, 2, 0), 13
D)( 3, -2, 0),
سؤال
Find the midpoint of the line segment joining the given points. (-5, 0, 2) and (-3, -2, 4)
A)( -4 ,-1 ,3 )
B)(
, 
, 
)
C)(
, 
, 
)
D)(
, 
, 
)
A)( -4 ,-1 ,3 )
B)(
, , )C)(
, , )D)(
, , ) سؤال
Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the xz-plane and 
units to the right of it.
units to the right of it. سؤال
Write an inequality to describe the region consisting of all points between (but not on) the
spheres of radius
and 
centered at the origin.
spheres of radius
and centered at the origin. سؤال
a.Find an equation of the sphere that passes through the point 
and has center 
.
b.Find the curve in which this sphere intersects the xy-plane.
and has center . b.Find the curve in which this sphere intersects the xy-plane.
سؤال
Sketch the plane in a three-dimensional space represented by the equation. z = 2
A)
B)
C)
D)
A)
B)
C)
D)
سؤال
Find the center and radius of the sphere. 
A)
B)
C)
D)
E)none of these
A)
B)
C)
D)
E)none of these
سؤال
Find the distance from 
to the xy-planes.
A)8
B)2
C)4
D)16
E)10
to the xy-planes.A)8
B)2
C)4
D)16
E)10
سؤال
Write inequalities to describe the solid rectangular box in the first octant bounded by the
planes
, 
, and 
.
planes
, , and . سؤال
Plot the given points in a three-dimensional coordinate system. (1, 2, 3)
A)
B)
C)
D)
A)
B)
C)
D)
سؤال
Find the length of the median of side AB of the triangle with vertices 
سؤال
Find an equation of the sphere with center 
that touches the xy-plane.
A)
B)
C)
D)
E)
that touches the xy-plane.A)
B)
C)
D)
E)
سؤال
سؤال
Suppose you start at the origin, move along the x-axis a distance of 
units in the positive direction, and then move downward a distance of 
units. What are the coordinates of your position?
A)
B)
C)
D)
E)
units in the positive direction, and then move downward a distance of units. What are the coordinates of your position?A)
B)
C)
D)
E)
سؤال
Find that the midpoint of the line segment from 
to 
.
A)
B)
C)
D)
E)
to .A)
B)
C)
D)
E)
سؤال
Write inequalities to describe the solid upper hemisphere of the sphere of radius 
centered at the origin.
centered at the origin. سؤال
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العب
ملء الشاشة (f)
Deck 12: Vectors and the Geometry of Space
1
Find an equation of the set of all points equidistant from the points and
A)
B)
C)
D)
E)
and A)
B)
C)
D)
E)
2
Reduce the equation to one of the standard forms.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
3
Find the point at which the line given by the parametric equations below intersects the plane.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
4
Find an equation of the plane that passes through the line of intersection of the planes and is perpendicular to the plane
and is perpendicular to the plane فتح الحزمة
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فتح الحزمة
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5
Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is times the distance from P to the yz-plane.
A)
B)
C)
D)
E)
times the distance from P to the yz-plane.A)
B)
C)
D)
E)
فتح الحزمة
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6
Find parametric equations for the line through and
and فتح الحزمة
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7
Find an equation for the surface obtained by rotating the parabola about the y-axis.
A)
B)
C)
D)
E)
about the y-axis.A)
B)
C)
D)
E)
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8
Find an equation for the surface obtained by rotating the line about the x-axis.
about the x-axis. فتح الحزمة
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9
Find the distance between the planes.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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10
Find an equation of the plane with x-intercept , y-intercept and z-intercept .
A)
B)
C)
D)
E)
, y-intercept and z-intercept .A)
B)
C)
D)
E)
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11
Find an equation for the surface consisting of all points that are equidistant from the point and the plane .
A)
B)
C)
D)
E)
and the plane .A)
B)
C)
D)
E)
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12
Identify the planes that are perpendicular.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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13
Find, correct to the nearest degree, the angle between the planes.
A)
B)
C)
D)
E)none of these
A)
B)
C)
D)
E)none of these
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14
Find the distance (correct to two decimal places) between the given parallel planes.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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15
Find parametric equations for the line through the point that is parallel to the plane and perpendicular to the line
A)
B)
C)
D)
E)
that is parallel to the plane and perpendicular to the line A)
B)
C)
D)
E)
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16
Find the equation of the line through and perpendicular to the plane
and perpendicular to the plane فتح الحزمة
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17
Which of the given lines is parallel to the line
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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18
Classify the surface.
A)A circular paraboloid with vertex and axis the z-axis.
B)A hyperboloid of one sheet with center and axis parallel to the z-axis.
C)A cone with axis parallel to the z-axis and vertex .
A)A circular paraboloid with vertex
and axis the z-axis.B)A hyperboloid of one sheet with center
and axis parallel to the z-axis.C)A cone with axis parallel to the z-axis and vertex
. فتح الحزمة
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19
An ellipsoid is created by rotating the ellipse about the x-axis. Find the equation of the ellipsoid.
A)
B)
C)
D)
E)
about the x-axis. Find the equation of the ellipsoid.A)
B)
C)
D)
E)
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20
Find the distance between the point and the plane Round your answer to two decimal place.
A)
B)
C)
D)
E)
and the plane Round your answer to two decimal place.A)
B)
C)
D)
E)
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21
A woman walks due west on the deck of a ship at mi/h. The ship is moving north at a speed of mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.
mi/h. The ship is moving north at a speed of mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth. فتح الحزمة
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22
Find the point of intersection.
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23
Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.
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24
Find the values of x such that the vectors and are orthogonal.
A)
B)
C)
D)
E)
and are orthogonal.A)
B)
C)
D)
E)
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25
Let and let u be a vector with length that starts at the origin and rotates in the xy - plane. Find the maximum value of the length of the vector .
A)
B)
C)
D)
E)
and let u be a vector with length that starts at the origin and rotates in the xy - plane. Find the maximum value of the length of the vector .A)
B)
C)
D)
E)
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26
Find the work done by a force that moves an object from the point to the point along a straight line. The distance is measured in meters and the force in newtons.
A)
B)
C)
D)
E)
that moves an object from the point to the point along a straight line. The distance is measured in meters and the force in newtons.A)
B)
C)
D)
E)
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27
Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as shown. Let ft, ft, , lb.
ft, ft, , lb. فتح الحزمة
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28
Find correct to three decimal places where , , .
A)
B)
C)
D)
E)
correct to three decimal places where , , .A)
B)
C)
D)
E)
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29
The tension T at each end of the chain has magnitude N and makes an angle with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth.
N and makes an angle with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth. فتح الحزمة
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30
Find an equation of the plane through the origin and parallel to the plane
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31
Find the cross product .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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32
Find parametric equations for the line through and parallel to the vector
and parallel to the vector فتح الحزمة
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33
A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is cm long.
Find the magnitude of the torque about P correct to two decimal places.
Let N, , .
cm long.Find the magnitude of the torque about P correct to two decimal places.
Let
N, , . فتح الحزمة
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34
Two forces with magnitudes 8 lb and 12 lb act on an object at a point P as shown in the figure. Find the magnitude of the resultant force F acting at P. Round the result to the nearest tenth.
A)
B)
C)
D)
E)
with magnitudes 8 lb and 12 lb act on an object at a point P as shown in the figure. Find the magnitude of the resultant force F acting at P. Round the result to the nearest tenth. A)
B)
C)
D)
E)
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35
Find a nonzero vector orthogonal to the plane through the points P, Q, and R.
A)
B)
C)
D)
E)None of these
A)
B)
C)
D)
E)None of these
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36
Find the unit vectors that are parallel to the tangent line to the curve at the point .
A)
B)
C)
D)
E)
at the point .A)
B)
C)
D)
E)
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37
Velocities have both direction and magnitude and thus are vectors. The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N W at a speed of km/h. (This means that the direction from which the wind blows is west of the northerly direction.) A pilot is steering a plane in the direction N E at an airspeed (speed in still air) of km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the ground speed of the plane. Round the result to the nearest hundredth.
W at a speed of km/h. (This means that the direction from which the wind blows is west of the northerly direction.) A pilot is steering a plane in the direction N E at an airspeed (speed in still air) of km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the ground speed of the plane. Round the result to the nearest hundredth. فتح الحزمة
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38
Calculate the given quantities if
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39
Calculate the angle between a and b (correct to the nearest degree).
A)
B)
C)
D)
E)None of these
A)
B)
C)
D)
E)None of these
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40
Find an equation of the plane that passes through the point and contains the line
and contains the line فتح الحزمة
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41
Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. A (-1, 0, 1), B (1, 1, -1), C (1, 1, 1)
A)1, 1, , both
B)3, 2, , right
C)1, 2, , neither
D)1, 1, , isosceles
A)1, 1,
, bothB)3, 2,
, rightC)1, 2,
, neitherD)1, 1,
, isosceles فتح الحزمة
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42
Draw a rectangular box with the origin and as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.
as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box. فتح الحزمة
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43
Find an equation of the sphere that passes through the point and has center .
and has center . فتح الحزمة
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44
Find the center and the radius of the sphere that has the given equation. + + - 6x + 4y = 0
A)( -3, 2, 0),
B)( 3, -2, 0), 13
C)( -3, 2, 0), 13
D)( 3, -2, 0),
+ + - 6x + 4y = 0A)( -3, 2, 0),
B)( 3, -2, 0), 13
C)( -3, 2, 0), 13
D)( 3, -2, 0),
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45
Find the midpoint of the line segment joining the given points. (-5, 0, 2) and (-3, -2, 4)
A)( -4 ,-1 ,3 )
B)( , , )
C)( , , )
D)( , , )
A)( -4 ,-1 ,3 )
B)(
, , )C)(
, , )D)(
, , ) فتح الحزمة
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46
Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the xz-plane and units to the right of it.
units to the right of it. فتح الحزمة
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47
Write an inequality to describe the region consisting of all points between (but not on) the
spheres of radius and centered at the origin.
spheres of radius
and centered at the origin. فتح الحزمة
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48
a.Find an equation of the sphere that passes through the point and has center .
b.Find the curve in which this sphere intersects the xy-plane.
and has center . b.Find the curve in which this sphere intersects the xy-plane.
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49
Sketch the plane in a three-dimensional space represented by the equation. z = 2
A)
B)
C)
D)
A)
B)
C)
D)
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50
Find the center and radius of the sphere.
A)
B)
C)
D)
E)none of these
A)
B)
C)
D)
E)none of these
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51
Find the distance from to the xy-planes.
A)8
B)2
C)4
D)16
E)10
to the xy-planes.A)8
B)2
C)4
D)16
E)10
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52
Write inequalities to describe the solid rectangular box in the first octant bounded by the
planes , , and .
planes
, , and . فتح الحزمة
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53
Plot the given points in a three-dimensional coordinate system. (1, 2, 3)
A)
B)
C)
D)
A)
B)
C)
D)
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54
Find the length of the median of side AB of the triangle with vertices
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55
Find an equation of the sphere with center that touches the xy-plane.
A)
B)
C)
D)
E)
that touches the xy-plane.A)
B)
C)
D)
E)
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56
Find the standard equation of the sphere with center C and radius r. C (3, -5, 3); r = 7
A)(x - 3)2 + (y + 5)2 + (z - 3)2 = 7
B)(x - 3)2 + (y + 5)2 + (z - 3)2 = 49
C)(x + 3)2 + (y - 5)2 + (z + 3)2 = 7
D)(x + 3)2 + (y - 5)2 + (z + 3)2 = 49
A)(x - 3)2 + (y + 5)2 + (z - 3)2 = 7
B)(x - 3)2 + (y + 5)2 + (z - 3)2 = 49
C)(x + 3)2 + (y - 5)2 + (z + 3)2 = 7
D)(x + 3)2 + (y - 5)2 + (z + 3)2 = 49
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57
Suppose you start at the origin, move along the x-axis a distance of units in the positive direction, and then move downward a distance of units. What are the coordinates of your position?
A)
B)
C)
D)
E)
units in the positive direction, and then move downward a distance of units. What are the coordinates of your position?A)
B)
C)
D)
E)
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58
Find that the midpoint of the line segment from to .
A)
B)
C)
D)
E)
to .A)
B)
C)
D)
E)
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59
Write inequalities to describe the solid upper hemisphere of the sphere of radius centered at the origin.
centered at the origin. فتح الحزمة
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60
Determine whether the given points are collinear. A (-3, -2, -3), B (-9, -5, 0), and C (-1, -1, -4)
A)Collinear
B)Not collinear
A)Collinear
B)Not collinear
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