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					Math 233                                       Vectors!                                      Version: 8-29-05

Vector Factoids:
      Know The Difference!
             Scalar: a quantity represented by a real number
             Vector: a quantity represented by both a direction and a magnitude
      Displacement Vector: measures the displacement between two points
      A=(2, 3) and B=(5,2), then AB  5  2, 2  3  3,  1 Always remember: Head – Tail or To – From

       Magnitude: the magnitude of a vector a, b, c is:        a 2  b2  c 2
       Unit Vector: a vector whose magnitude is 1 (so it measures direction, or relative slope
       only!)
                    
       Notation: b  b  4,2  4i  2 ˆ where iˆ and ˆ are unit vectors in the positive x-
                                ˆ     j              j
       and y- axis directions.
       More Notation: unit vectors are frequently given “hats” (or carets) to denote their
       unique property ( iˆ ).
       Equality: Two vectors are equal if they have the same length and direction

Some Practice:
   1. Find the displacement from the point P1 to P2 for P  4,6
                                                         1                 P2  7,10 :




                                                                                   
   2. Given the vectors u and v shown, sketch the vectors 2u , u  v , and u  v .




                                      
                                      v


                                           
                                           u



                                                                                   
3. Two adjacent sides of a regular hexagon are given as the vectors u and v . Label the
                                                           
   remaining sides in terms of u and v (for example,  u or u  v ).




                                                          
                                                          v
                                          
                                          u
                        
4. Given the vector v  2,4 .
                                                                         
   a) What is the magnitude of v (noted as v or sometimes v )?
       (Hint: draw a picture and then use the Pythagorean Theorem)




                                                    
   b) Find a unit vector in the direction of v .
                                                                               
       (Hint: the magnitude of a unit vector is 1… what was the magnitude of   v   ?)




5. A vector has length 8 and makes a 30-degree angle with the x-axis. What are the values
   of the iˆ and ˆ components?
                 j




                                                ˆ
6. The velocity of a truck is determined to be 9i  11 ˆ . If another truck is traveling twice as
                                                       j
   fast in the opposite direction, what is the velocity vector of the second truck? What is
   the speed of the second truck (note: speed is a scalar, velocity is a vector)?




7. Super Bonus Problem: An airplane is flying at an airspeed of 600 km/hr in a cross-wind
   that is blowing from the northeast at a speed of 50 km/hr. In what direction should the
   plane head to end up going due east? What will the groundspeed end up being (this is the
   speed in the resultant, or due east, direction)?

				
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posted:12/11/2011
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