Deck 10: Systems of Equations and Inequalities

Question 1

Express the given vector in terms of the unit vectors i, j, and k.

Answer 1

Question 2

Find the distance between the points
and
.

Answer 2

Question 3

Determine whether
is orthogonal to
. Justify your answer.

Answer 3

, orthogonal

Question 4

Given
and
, calculate
and then resolve
into
and
.

Question 5

Find
.

Question 6

Given that the forces $\mathbf { F } _ { 1 } = ( - 10,3 )$ , $\mathbf { F } _ { 2 } = ( - 4,1 )$ and $\mathbf{F}_{3}=\langle 4,-10\rangle$ are acting on a point $p$ , find the resultant force, magnitude and the additional force required in order for the forces to be in equilibrium.

Question 7

A water tank is in the shape of a sphere of radius 10 feet. The tank is supported on a metal circle 8 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Question 8

Let u be the vector with initial point
and terminal point
. Express u in terms of i and j.

Question 9

A jet heads in the direction N
E at a speed of
mi/h. If the jet experiences a
mi/h crosswind flowing due east, find (a) the true velocity
of the jet
b) the true speed of the jet
c) the direction of the jet.

Question 10

Find
and
, given that
and
.

Question 11

Find the angle between
and
.

Question 12

If the vector v has initial point P, what is its terminal point?

Question 13

Two vectors u and v are given. Find the angle to one decimal place (expressed in degrees) between u and v.

Question 14

Find the magnitude and direction (in degrees) of the vector
.

Question 15

Find the vector
having length
and direction
.

Question 16

Find the work done by the force
in moving an object from
to
.

Question 17

A man pulls a sled horizontally by exerting
on the rope that's tied to its front end. If the rope makes an angle of
with horizontal, find the work done in moving the sled
ft.

Question 18

Find an equation of a sphere with the given radius r and center C.
;

Question 19

Find the component of
along
if
and
.

Question 20

Find the direction of the vector $- 2 \mathbf { i } + 2 \sqrt { 3 } \mathbf { j }$
.

Question 21

A plane has normal vector n and passes through the point P. Find an equation for the plane.

Question 22

The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$
. $| \mathbf { a } | = 4 , \mathbf { b } \mid = 5 , \theta = 60 ^ { \circ }$

Question 23

Find the area of the parallelogram determined by the given vectors.

Question 24

A jet heads in the direction N $30 ^ { \circ }$ E at a speed of $400$ mi/h. If the jet experiences a $20$ mi/h crosswind flowing due east, find
a) the true velocity w
of the jet
b) the true speed of the jet
c) the direction of the jet.

Question 25

Let u be the vector with initial point
and terminal point
. Express u in terms of i and j.

Question 26

Find the work done by the force
in moving an object from
to
.

Question 27

Find the magnitude and direction (in degrees) of the vector
.

Question 28

Determine whether
is orthogonal to
. Justify your answer.

Question 29

Find parametric equations for the line that passes through the point P and is parallel to the vector v.

Question 30

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$

A) $C ( 2,2,4 ) ; r = \sqrt { 2 }$
B) $C ( 0,2,4 ) ; r = 4$
C) $C ( 2,4,0 ) ; r= \sqrt { 2 }$
D) $C ( 2,4,0 ) ;r= 2$
E)none of these

Question 31

Two vectors a and b are given. (a) Find a vector perpendicular to both a and
b
b)Find a unit vector perpendicular to both a and
b.

Question 32

Find the component of
along
if
and
.

Question 33

Find the angle between
and
.

Question 34

Given
and
, calculate
and then resolve
into
and
.

Question 35

Given that the forces $\mathbf { F } _ { 1 } = ( - 10,3 )$ , $\mathbf { F } _ { 2 } = ( - 4,1 )$ and $\mathbf{F}_{\mathrm{3}}=\langle 4,-10\rangle$
are acting on a point $p$ , find the resultant force, magnitude and the additional force required in order for the forces to be in equilibrium.

Question 36

Find the distance between the points
and
.

Question 37

Find the angle between $u$ and $\mathbf { Y }$ .
$\mathbf { u } = ( - 5,3 ) , \mathbf { v } = \langle \sqrt { 2 } , 7 )$

Question 38

Find the direction of the vector
.

Question 39

Find the vector
having length
, and direction
.

Question 40

Find
.

Question 41

A water tank is in the shape of a sphere of radius 15 feet. The tank is supported on a metal circle 12 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Question 42

$\mathbf { u } = \left( - \frac { 1 } { 2 } , 2 \right\rangle$ is orthogonal to $\mathbf { v } = \left( - 2 , \frac { 1 } { 2 } \right)$

Question 43

Find the area of the parallelogram determined by the given vectors.

Question 44

Find the angle between and
)

A)
B)
C)
D)
E)

Question 45

If the vector v has initial point P, what is its terminal point?

Question 46

A plane has normal vector n and passes through the point P. Find an equation for the plane.

Question 47

Find an equation of a sphere with the given radius r and center C.
;

Question 48

Two vectors u and v are given. Find the angle correct to one decimal place (expressed in degrees) between u and v.

Question 49

Let u be the vector with initial point and terminal point
) Express u in terms of i and j

A)
B)
C)
D)
E)

Question 50

A jet heads in the direction N $30 ^ { \circ }$ E at a speed of $400$ mi/h. If the jet experiences a $20$ mi/h crosswind flowing due east, find the true velocity w

A) $w = 200 \mathbf { i } + 420 \sqrt { 3 } \mathbf { j }$
B) $w = 420 \mathbf { i } + 200 \sqrt { 3 } \mathbf { j }$
C) $w = 200 \mathbf { i } + 220 \mathbf { j }$
D) $w = 220 \mathbf { i } + 200 \sqrt { 3 } \mathbf { j }$
E)none

Question 51

Find parametric equations for the line that passes through the points P and Q.

Question 52

Given that the forces ,
And
Are acting on a point
, find the resultant force

A)
B)
C)
D)
E)

Question 53

The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$
. Write the answer correct to three decimal places. $| \mathbf { a } | = 0.12 , \mathbf { b } \mid = 1.25 , \theta = 85 ^ { \circ }$

Question 54

Find the magnitude of the vector

A)
B)
C)
D)
E)

Question 55

Find and
, given that
And

A)
,
B)
,
C)
,
D)
,
E)none

Question 56

Find a vector that is perpendicular to the plane passing through the three given points.

Question 57

Express the given vector in terms of the unit vectors i, j, and k.

Question 58

Find the direction of the vector

A)
B)
C)
D)
E)

Question 59

Find the vector having length
, and direction

A)
B)
C)
D)
E)

Question 60

Find .

A)
B)
C)
D)
E)

Question 61

Find the magnitude of the vector

A)
B)
C)
D)
E)

Question 62

Given $\mathbf { u } = \langle 2,1 \rangle$ and $\mathbf { v } = \langle - 3,1 \rangle$ , calculate $\operatorname { proj } _ { \mathbf { x } } \mathbf { u }$

A) $( 3 , - 2 )$
B) $( 3 , - 1 )$
C) $\left( \frac { 2 } { 3 } , - \frac { 1 } { 2 } \right)$
D) $\left( - \frac { 3 } { 2 } , \frac { 1 } { 2 } \right)$
E) $\left( \frac { 3 } { 2 } , - \frac { 1 } { 2 } \right)$

Question 63

Describe the surface represented by the given equation.

A) Plane parallel to the xy-plane
B) Plane parallel to the xz-plane
C) Sphere with radius 7
D) Sphere with radius
E) none of these

Question 64

A water tank is in the shape of a sphere of radius 10 feet. The tank is supported on a metal circle 8 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle

A)
B)
C)
D)
E) none of these

Question 65

Find the distance between the points and

A)83
B)
C)
D)
E) 9

Question 66

Let u be the vector with initial point and terminal point
) Express u in terms of i and j

A)
B)
C)
D)
E)none

Question 67

Find the work done by the force in moving an object from
To

A)
B)
C)
D)
E) none

Question 68

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v.

A)
B)
C)
D)
E) none of these

Question 69

Find an equation of the plane that passes through the points P, Q, and R.

A)
B)
C)
D)
E) none of these

Question 70

A man pulls a sled horizontally by exerting on the rope that is tied to its front end. If the rope makes an angle of
With horizontal, find the work done in moving the sled
Ft

A)
Ft-lb
B)
Ft-lb
C)
Ft-lb
D)
Ft-lb
E)
Ft-lb

Question 71

Express the given vector in terms of the unit vectors i, j, and k.

A)
B)
C)
D)
E) none of these

Question 72

Find the vector having length
, and direction

A)
B)
C)
D)
E)

Question 73

The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$
. $| \mathbf { a } | = 0.12 , \mathbf { b } \mid = 1.25 , \theta = 85 ^ { \circ }$

Question 74

Find the component of $u$ along $\mathbf { Y }$
If $\mathbf { u } = ( - 6,7 )$ and $v = ( 3,4 )$

A) $2$
B) $5$
C) $\frac { \sqrt { 10 } } { 7 }$
D) $\frac { \sqrt { 5 } } { 2 }$
E) none

Question 75

Find the area of the parallelogram determined by the given vectors.

A)
B)
C)
D)
E) none of these

Question 76

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$

A) $C ( 2,2,4 ) ; r = \sqrt { 2 }$
B) $C ( 0,2,4 ) ; r = 4$
C) $C ( 2,4,0 ) ; r = \sqrt { 2 }$
D) $C ( 2,4,0 ) ; r = 2$
E)none of these

Question 77

If the vector v has initial point P, what is its terminal point?

A)
B)
C)
D)
E) none of these

Question 78

Find and
, given that
And

A)
,
B)
,
C)
,
D)
,
E)none

Question 79

Find parametric equations for the line that passes through the point P and is parallel to the vector v.

A)
B)
C)
D)
E) none of these

Question 80

The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . $| \mathbf { a } | = 4 , \mathbf { b } \mid = 5 , \theta = 60 ^ { \circ }$

A) $20 \sqrt { 3 }$
B) $10 \sqrt { 2 }$
C) $10 \sqrt { 3 }$
D) $10$
E) none of these