1. You are given the following points: A = (13,2,-17), B = (19,0,-8), C = (-9,-3

Question

1. You are given the following points: A = (13,2,-17), B = (19,0,-8), C = (-9,-3,16)

Which point is closest to the yz-plane???

What is the distance from the yz-plane to this point????

  Which point is farthest from the xy-plane?????

  What is the distance from the xy-plane to this point?????

  Which point lies on the xz-plane?????

2. Find an equation of the sphere that passes through the origin and whose center is (9, -9, 3).

__________________________________________ = 0

Note that you must put everything on the left hand side of the equation and that we desire the coefficients of the quadratic terms to be 1.

3. 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 18 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

4. Which is traveling faster, a car whose velocity vector is 21i? +26j? , or a car whose velocity vector is 30i? , assuming that the units are the same for both directions?

At what speed is the faster car traveling?

5. The unit vector oppositely directed to i?5j+k is ____________________

The unit vector that has the same direction as the vector from the point A=(1,?4) to the point B=(?1,?3) is ______________

Explanation / Answer

1)

a)

for YZ-plane to find the closest point check the magnitude of x-coordinate int the given points

we see that  (-9,-3,16) is the closest point to YZ-plane

b)

the distance is magnitude of X-coordniate = 9

c)

for the XY plane check the Z-coordinate

we see that A = (13,2,-17), is at the farthest

d)

the distace is the magnitude of z cordinate

= 17

e)

if a point lies on xz plane the y-cordinate = 0

we see that B = (19,0,-8) is that point

2)

since the sphere passes through the origin and the center of the sphere is (9, -9, 3) the radius is the distace between the origin and (9, -9, 3)

= sqrt [ (9)^2 + (-9)^2 +(3)^2 ]

=sqrt(171)

so the equation of the sphere is as follows

(x-9)^2 + (y+9)^2 +(z-3)^2 = (sqrt(171)^2

(x-9)^2 + (y+9)^2 +(z-3)^2 = 171

(x-9)^2 + (y+9)^2 +(z-3)^2 -171 =0

3)

he ship moves north 18 mph
The child moves East 1 mph

The two vectors make the two short sides of a right-angle triangle.
The resultant = sqrt( 18^2 + 1^2 ) =sqrt(325) =18.0277mph

find the direction

angle measured northward from east, is the arc-tangent of (18/1)

=86.82 degrees

=0.4823*pi radians

=1.515 radians

and speed = 18.0277 mph

4)

magnitude of car with velocity  21i +26j

=sqrt(21^2 + 26^2 ) = sqrt(1117)

=33.4215 m/s

which is greater than 30 m/s

so the car with  21i +26j velocity is faster

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