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Calc 3 10.2

 

<== Calculus 3
Number Question Answer
See the work

1.

                     

 

 

Let a = <3,0,5> and b = <-3,4,-5>       
Compute:
a+b = <__,__,__>
a-b = <__,__,__>
2a = <__,__,__>
3a+4b = <__,__,__>
lal= __________

  a+b = <0,4,0> 
  a-b = <6,-4,10> 
  2a = <6,0,10> 
  3a+4b = <-3,16,-5> 
  lal= 5.83095194845  

The work

Point Cost: 1

                         

2.

Let a = < 5, 5, -5 >.
Find a unit vector in the same direction as a.
< , , >

< , , >
The work

Point Cost: 1
3.

Find the unit vector in the direction opposite to v=<1,-3>.


<-0.316228,0.948683>
The work

Point Cost: 1
4.

Find the components and length of the following vectors:
4i + 4j
Components: < ,>
Length:

4i - 1j
Components: < ,>
Length:

-5i - 1j
Components: < ,>
Length:

-1i - 3j
Components: < ,>
Length:

4i + 4j

Components: < 4, 4 >
Length: sqrt(32)

4i - 1j

Components: < 4, -1 >
Length: sqrt(17)

-5i - 1j

Components: < -5, -1 >
Length: sqrt(26)

-1i - 3j

Components: < -1, -3 >
Length: sqrt(10)
The work

Point Cost: 1
5.

If P=(3,4) and Q=(7,7), find the components of vec{PQ}
vec{PQ}=

< 4, 3 >
The work

Point Cost: 1
6.

Let R=(5,1). Find the point P such that vec{PR} has components <1,-1>.

( 4, 2 )
The work

Point Cost: 1
7. What is the terminal point of the vector a=<2,1> based at P=(3,5)?
( 5, 6 )
The work

Point Cost: 2
8.

 

If a= <-1,-5,-3>, find a unit vector in the
same direction as a=<__,__,__>

 

    a=<-.16903,-.84515,-.5070926>
The work

Point Cost: 1
9.
Find a vector a that has the same direction as < -6, 7, 6 > but has length 3. a=

a=<-1.63637,1.90909,1.63637>
The work

Point Cost: 2

10.

 

 

 

a=<8,0,7> and b=<-3,2,-7> are vectors.
Compute the requested following vectors:   
A. 6a= (__,__,__)
B. a-b= (__,__,__)
C. lal=__________
D. a+b= (__,__,__)

 

   A. 6a= <48,0,42>
   B. a-b= <11,-2,14>
   C. lal= 10.630145812734 
   D. a+b= <5,2,0>
The work

Point Cost: 1

11.

 

 

 

 

A person walks due east on the front of a ship
at 3 miles per hour. If the ship moves
north at a speed of 12 miles per hour,
find the direction and speed of the person
relative to the water surface
Speed=__________mph
The angle of the direction from north=
______radians

 

   Speed=12.3693169 mph
  

   The angle of the direction
   from north= .24498 radians

The work

Point Cost: 3

12.

 

 

A horizontal clothesline is tied from 2 poles,
16 meters apart. If a 4 kilogram mass is placed
in the middle of the line, and it sags 1 meter
What is the magnitude of the tension at the
ends of the clothesline? T=_______N

 

   T=158.0202525N
The work

Point Cost: 4

13.

 

 

Nine ring wraiths fly from point a to point b.
point b is directly north of point a. Wind is
coming from the west at 55 mph. To travel in
a straight line the ring wraiths decide to head NW.   
What speed should they fly?

 

   77.78174593
The work

Point Cost: 3
14. Let a=<5,5> and b=<1, -4>
Show that there are scalars s and t so that
sa+tb=<-5,-30>
s=________
t=________

    s=-2

    t=5

The work

Point Cost: 3
15. A child walks due east on the deck of a ship at 1 miles per hour.
The ship is moving north at a speed of 6 miles per hour.

Find the speed and direction of the child relative to the surface of the water.

Speed = ____ mph

The angle of the direction from the north = _______ (radians)
Speed = 6.083 mph

The angle from the north = 0.165 (radians)
The work
16. In the figure below the y-axis points north,
the x-axis points east, and the xy-plane
corresponds to the surface of the water.
Suppose a boat is at point B, a submarine is 8 units
below point S, and a helicopter is 11 units above point H.
  1. Find the displacement vector and the distance from the submarine to the boat.
    Displacement:
    Distance:

  2. Find the displacement vector and distance from the helicopter to the boat.
    Displacement:
    Distance:

  3. Find the displacement vector and distance from the submarine to the helicopter.
    Displacement:
    Distance:
  1. Displacement: < -2, 3, 8 >
    Distance: 8.77496

  2. Displacement: < 2, 2, -11 >
    Distance: 11.3578

  3. Displacement: < -4, 1, 19 >
    Distance: 19.4422
The work

Point Cost: 2