PN α ( .24f) Equation ( .24a) gives the magnitude of the resultant and Eqs. ( .24e) and ( .24f) its direction. Equation ( .24a) is known as the Law of cosines and Eq. ( .24d) as the Law of sines .
⊳ Example . A motorboat is racing towards north at km/h and the water current in that region is km/h in the direction of ° east of south. Find the resultant velocity of the boat. Answer The vector v b representing the velocity of the motorboat and the vector v c representing the water current are shown in Fig.
. in directions specified by the problem. Using the parallelogram method of addition, the resultant R is obtained in the direction shown in the figure. Fig.
. We can obtain the magnitude of R using the Law of cosine : v v cos120 - / km/h ) ≅ To obtain the direction, we apply the Law of sines v c φ or, sin φ c sin sin120 . . .
≅ φ ≅ . . MOTION IN A PLANE In this section we shall see how to describe motion in two dimensions using vectors. .
. Position Vector and Displacement The position vector r of a particle P located in a plane with reference to the origin of an x-y reference frame (Fig. . ) is given by where x and y are components of r along x -, and y - axes or simply they are the coordinates of the object.
(a) (b) Fig. . (a) Position vector r . (b) Displacement ∆ r and average velocity v of a particle.
Suppose a particle moves along the curve shown by the thick line and is at P at time t and P ′ at time t ′ [Fig. . (b)]. Then, the displacement is : ∆ r = r ′ – r ( .
) and is directed from P to P ′ . We can write Eq. ( . ) in a component form: