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Calc 3 10.3

 

<== Calculus 3

 

Number Question Answer
See the work

        1.         

 

If a = <6,0,3> and b = <-4,2,3>,   
find a • b =________.

 

-15
The work

Point Cost: 1

2.

 

 


If u = <-5,5,1> and v= <6,2,-8>
Compute 4llull=______
Compute llvll=______
Compute u•v =______
Compute llull= 7.14143
Compute llvll= 10.19804 
Compute u•v = -28
The work

Point Cost: 1

3.





Find a • b if
lal = 7,
lbl = 4,
and the angle between a and b
is π/-10 radians.
a•b=_________

 

a•b= 26.62958 
The work

Point Cost: 2

4.

 

 

Find a•b if
a= <-2,-5,-3> and b= <1,5,-5> 
Is the angle between the vectors ”acute”,    
”obtuse” or ”right”?

 

-12

Obtuse

The work

Point Cost: 2

5.

 

 

Find ab if
a= <-3,-2,-2> and b= <0,1,0> and
Is the angle between the vectors ”acute”,
”obtuse” or ”right”?

 

-2

Obtuse

 

The work

Point Cost: 2

6.

 

 

Find a vector orthogonal to both
< -5, 3, 0 > and to < 0, 3, 3 > of the form

< 1, ____, _____ >
< 1, 5/3, -5/3 >
The work

Point Cost: 3

7.

 

 

Assume that u \cdotp v = 9, \| u \| = 7, and \| v \| = 9.

What is the value of 8 u \cdotp ( 7 u  - 7 v )?

2240
The work

Point Cost: 4

8.

 

 

v = i+5j+9k

w = -i-5j+5k

Compute the dot product v dot w.

19
The work

Point Cost: 1

9.

 

 

The force on an object is  F = -17 j. For the vector  v = -5 i -4 j,
find:

(a) The component of F parallel to v:

(b) The component of F perpendicular to v:

The work, W, done by force F through displacement v:

(a): -8.29268 i -6.63415 j

(b): 8.29268 i-10.3659 j

W = 68
The work

Point Cost: 3

10.

 

 

For what values of b are the vectors < -25, b, 9 and < b, b^2, b > orthogonal?

b =

b = 0, -4 4
The work

Point Cost: 3

11.

 

 

Perform the following operations on the vectors u = <-2,1,2>, v = <-2,-3,-5>, and w = <2,0,-3>.

u dot w =

(u dot v)u =

((w dot w)u) dot u =

u dot v + v dot w =


u dot w = -10

(u dot v)u = < 18,-9,-18 >

((w dot w)u) dot u = 117

u dot v + v dot w = 2
The work

Point Cost: 3

12.

 

 

 

Determine if the pairs of vectors below are    
”parallel”, ”orthogonal”, or ”neither”.
a= <5,2,1> and b= <-15,-6,87> are
a= <5,2,1> and b= <-15,-6,-3> are
a= <5,2,1> and b= <15,6,2> are

 

Orthogonal
Parallel
Neither

 

The work

Point Cost: 2

13.

 

What is the angle in radians between
the vectors a= <-9,-5,-6> and b= <-7,7,-2>?  

 

1.2319859

 

The work

Point Cost: 2

14.

 

 

A rectangular box has length 19 inches,
width 7 inches, and a height of 11 inches.
Find the angle between the diagonal of the
base and the diagonal of the left hand side

 

Angle in radians: 1.3841131
Angle in degrees: 79.303984

 

The work

Point Cost: 5

15.

 

Find the scalar and vector projection of the
vector b=<4,5,-1> onto the vector a= <3,1,-1>.

 

Scalar projection: 5.427204202
Vector projection: <54/11,18/11,-18/11>

 

The work

Point Cost: 3

16.

 

Find the scalar and vector projection of the
vector b= <1,1,-5> onto the vector a= <1,-3,1>.   

 

Scalar projection: -2.110579412
Vector projection <-7/11, 21/11, -7/11>

 

The work

17.

 

 

 

 

Gandalf the Grey started in a Forest
at a point with coordinates (-1, 3). He arrives
in these Hills at the point (0, 8). If he begins to
walk in the direction of the vector v= 3i+1j and
changes direction only once, when he turns at a
right angle, what are the coordinates of the point
where he makes the turn.

 

(7/5, 3.8)

 

The work

18.

 

 

 

A constant force F= 2i+3j+3k moves an object
along a straight line from the point (0, 1, 4)
to the point (0, -3,-4). Find the work done if
the distance is measured in meters and the force
in newtons.

 

-36 J

 

 

The work

19.

 

 

A constant force F= -4i+2j+7k moves an object along a
straight line from point (-1, -7, 7) to point (10, 0, -3).
Find the work done if the distance is measured in meters   
and the magnitude of the force is measured in newtons.

 

Work: -100 Nm

 

 

The work

20.

 

 

A woman exerts a horizontal force of 9 pounds
on a box as she pushes it up a ramp that is 7 feet
long and inclined at an angle of 30 degrees above
the horizontal. Find the work done on the box.

 

Work: 54.55960043 ft-lb

 

The work