Vectors in the Plane & Derivatives

Problem:

Find unit vectors tangent and normal to the parametrized curve

Answer:  
,  so find the tangent and normal at the point (2,4).
  slope of the tangent = 

  At t = 1, m = 4

  Tangent Vector:  <1, 4>  or any other vector in component form that indicates its slope is 4.
  Normal Vector:  <-4,1>  or any other vector in component form that indicates its slope is -1/4.

  These unit vectors are   and   , respectively.
 

Problem:

An airplane, flying in the direction 20o east of north at 325 mph in still air, encounters a 40-mph tail wind acting in the direction 40o west of north.  The airplane maintains its compass heading but, because of the wind, acquires a new ground speed and direction.  What are they?

The ground speed vector is the resultant of the airspeed vector and the windspeed vector.

airspeed = <325 cos 70o, 325 sin 70o> since 20o east of north is 70o in standard position.
windspeed = <40 cos 130o, 40 sin 130o> since 40o west of north is 130o in standard position.

ground speed = <325 cos 70o + 40 cos 130o, 325 sin 70o + 40 sin 130o >
                     =  <85.445, 336.042 >

The magnitude of the ground speed is \/(85.4452 + 336.0422) = 346.735 mph.
   (Note:  Use the original values in your calculator, not the rounded values)

The direction is obtained by finding