Vectors in mechanics

[ In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O.] Two ships, P and Q, are moving with constant velocities. The velocity of P is (9i – 2j)km h^–1 and the velocity of Q is (4i + 8j) km h^–1

Find the direction of motion of P, giving your answer as a bearing to the nearest degree.

When t = 0, the position vector of P is (9 i + 10 j)km and the position vector of Q is (i + 4 j) km. At time t hours, the position vectors of P and Q are p km and q km respectively.

Find an expression for

p in terms of t,

q in terms of t.

Hence show that, at time t hours,

Find the values of t when the ships are 10km apart.

[ In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O.] Two cars P and Q are moving on straight horizontal roads with constant velocities. The velocity of P is (15i + 20j) m s^–1 and the velocity of Q is (20i – 5j) m s^–1

Find the direction of motion of Q, giving your answer as a bearing to the nearest degree.

At time t = 0, the position vector of P is 400 i metres and the position vector of Q is 800 j metres. At time t seconds, the position vectors of P and Q are p metres and q metres respectively.

Find an expression for

p in terms of t,

q in terms of t.

Find the position vector of Q when Q is due west of P.

Two forces (4 i – 2 j) N and (2 i + q j) N act on a particle P of mass 1.5 kg. The resultant of these two forces is parallel to the vector (2 i + j).

Find the value of q.

At time t = 0, P is moving with velocity (–2i + 4j)m s^–1.

Find the speed of P at time t = 2 seconds.

[ In this question i and j are horizontal unit vectors due east and due north respectively. Position vectors are given with respect to a fixed origin O. ] A ship S is moving with constant velocity (3i + 3j) km h^–1. At time t = 0, the position vector of S is (–4 i + 2 j) km.

Find the position vector of S at time t hours.

A ship T is moving with constant velocity (–2i + nj) km h^–1. At time t = 0, the position vector of T is (6 i + j) km. The two ships meet at the point P.

Find the value of n.

Find the distance OP.

[ In this question, i and j are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin. ] A ship sets sail at 9 am from a port P and moves with constant velocity. The position vector of P is (4 i – 8 j) km. At 9.30 am the ship is at the point with position vector (i – 4 j) km.

Find the speed of the ship in km h^–1.

Show that the position vector r km of the ship, t hours after 9 am, is given by r = (4 – 6 t) i + (8 t – 8) j.

At 10 am, a passenger on the ship observes that a lighthouse L is due west of the ship. At 10.30 am, the passenger observes that L is now south-west of the ship.

Find the position vector of L.

[ In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin. ] A ship S is moving with constant velocity (–12i + 7.5j) km h^–1.

Find the direction in which S is moving, giving your answer as a bearing.

At time t hours after noon, the position vector of S is s km. When t = 0, s = 40 i – 6 j.

Write down s in terms of t.

A fixed beacon B is at the point with position vector (7 i + 12.5 j) km.

Find the distance of S from B when t = 3

Find the distance of S from B when S is due north of B.

[ In this question i and j are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin O.]

Find, to the nearest degree, the bearing on which Q is moving.

At 2 pm, ship P is at the point with position vector and ship Q is at the point

Write down expressions, in terms of t, for

Find the time when

Q is due north of P,

Q is north-west of P.

A particle P of mass 2 kg is moving under the action of a constant force F newtons. The velocity of P is (2i − 5j) m s¹ at time t = 0, and (7i +10j) m s¹ at time t 5 s. Find

the vector F in the form ai + bj,

the value of t when P is moving parallel to i.

[ In this question, i and j are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship S is moving along a straight line with constant velocity. At time t hours the position vector of S is s km. When t = 0, s = 9 i – 6 j. When t = 4, s = 21 i + 10 j. Find

the speed of S,

the direction in which S is moving, giving your answer as a bearing.

Show that s = (3 t + 9) i + (4 t – 6) j.

A lighthouse L is located at the point with position vector (18 i + 6 j) km. When t = T, the ship S is 10 km from L.

Find the possible values of T.

[ In this question i and j are horizontal unit vectors due east and due north respectively. ] A hiker H is walking with constant velocity (1.2i – 0.9j) m s^–1.

Find the speed of H.

C B

A horizontal field OABC is rectangular with OA due east and OC due north, as shown in Figure 3. At twelve noon hiker H is at the point Y with position vector 100 j m, relative to the fixed origin O.

Write down the position vector of H at time t seconds after noon.

At noon, another hiker K is at the point with position vector (9 i + 46 j) m. Hiker K is moving with constant velocity (0.75i + 1.8j) m s^–1.

Show that, at time t seconds after noon,

Hence,

show that the two hikers meet and find the position vector of the point where they meet.

[ In this question, the unit vectors i and j are due east and due north respectively. ] A particle P is moving with constant velocity (–5i + 8j) m s^–1. Find

the speed of P,

the direction of motion of P, giving your answer as a bearing.

At time t = 0, P is at the point A with position vector (7 i – 10 j) m relative to a fixed origin O. When t = 3 s, the velocity of P changes and it moves with velocity (ui + vj) m s^–1, where u and v are constants. After a further 4 s, it passes through O and continues to move with velocity (ui + vj) m s^–1.

Find the values of u and v.

Find the total time taken for P to move from A to a position which is due south of A.