Questions about Chapter 3: Vectors.

Q1. Vector A has a magnitude 6 cm and is at 36.9o counterclockwise from the + x-axis.  It is added to a vector B, and the resultant is a vector of magnitude 6 cm at 53.1o counterclockwise from the + y-axis.  Find the magnitude and direction of the vector B.

ANS:  9.60 cm along the - x-axis.

Q2.  Three displacement vectors are in the same plane and are expressed as: a = ( 4 i   -   j ) m ;  b = ( - 3 i + 2 j ) m ;  and  c = ( - 3 j ) m. What is the vector r where r = a - b +c ?  What angle does r make with the x-axis ?

ANS:  r = ( 7 i - 6 j ) m ;  - 40.6o

Q3.   Two vectors are given by

a = 2 i + 3 j + 4 k     and       b = i - 2 j + 3 k.

Calculate the vector    c = 3 a - 2 b.

ANS:    c = 4 i + 13 j + 6 k .

Q4. A car travels 30 km due south and then 40 km due west.  Find the magnitude and    direction of the resultant displacement of the car.  Assume no change in the   elevation between the initial and final positions.

ANS:  50 km ;   53.1o   west of south.

Q5. Two vectors L and M are defined by

L = ( 4 i - 8 j )     m   and    M = ( 8 i + 2 j )   m.

Find the magnitude and direction of the vector ( 2 L - M ).

ANS:  18.0 m at 270o with the + x-axis.  =  -18 j

Q6.  Two points A and B are located in the xy-plane.  The Cartesian co-ordinates of A are ( 1 , 4 ) and those of B are ( 4 , 1 ) , in m. Write, in unit vector notation, the vector R that goes from point A to point B.  What is the distance between points A and B ?

ANS:   R =  3 i - 3 j  ;   Distance = 4.24 m.

Q7  Starting from airport A , an airplane flies 300 km east, then 400 km northeast ( 45o   north of east ), and then 100 km at an angle of 30o west of north to arrive finally at airport B.

(a)  What is the direct distance between airport A and airport B ?

(b)  In what direction should a pilot of a direct flight from airport A to airport   B fly the airplane ?  ( Assume there is no wind.)

ANS: 648 km; 34.7o N of E.

Q8.  A person walks 12.0 km, 20o south of east, and then walks 15.0 km, 10o south of west.  Call these displacements A and B, respectively.

(a)  Write down in unit-vector notation these two displacements A and B. Take east   along the positive x-axis and north along the positive y-axis.

(b)  Find the magnitude and direction of the resultant vector.

ANS: A = ( 11.3 i  - 4.10 j ) km ;             B = ( - 14.8 i  - 2.61 j )  km

ANS: Magnitude = 7.57 km; Direction = 62.5o S of W .