Magnitude & direction of a vector

Given that

• PQ → = 2i + 8j – 2k

• QR → = 6i + 6k

find → PR.

Hence show that triangle PQR is both right-angled and isosceles.

Relative to a fixed origin O

• the point A has position vector 5 i + 3 j + 2 k

• the point B has position vector 2 i + 4 j + a k

where a is a positive integer.

Find the smallest value of a for which

Figure 1 shows a sketch of triangle ABC.

Given that

find AC

show that cosABC =

Figure 1 shows a sketch of triangle ABC. Given that

find AC

show that cosABC =

With respect to a fixed origin, the point A with position vector i + 2 j + 3 k lies on the line l₁ with equation 1 0

and the point B with position vector 4i + pj + 3k, where p is a constant, lies on the line l₂ with equation 7 3

Find the value of the constant p.

Show that l₁ and l₂ intersect and find the position vector of their point of intersection, C.

Find the size of the angle ACB, giving your answer in degrees to 3 significant figures.

Find the area of the triangle ABC, giving your answer to 3 significant figures.

The line l₁ has vector equation

and the line l₂ has vector equation

where λ and μ are parameters. The lines l₁ and l₂ intersect at the point A and the acute angle between l₁ and l₂ is .

Write down the coordinates of A.

Find the value of cos .

The point X lies on l₁ where λ = 4.

Find the coordinates of X.

Hence, or otherwise, show that = 4√26.4 26.

The point Y lies on l₂. Given that the vector is perpendicular to l₁,

find the length of AY, giving your answer to 3 significant figures.

[ In this question i and j are horizontal unit vectors due east and due north respectively ] Two forces F₁ and F₂ act on a particle P of mass 0.5 kg. F₁ = (4i – 6j) N and F₂ = (pi + qj) N. Given that the resultant force of F₁ and F₂ is in the same direction as –2i – j,

show that p – 2 q = –16

Given that q = 3

find the magnitude of the acceleration of P,

find the direction of the acceleration of P, giving your answer as a bearing to the nearest degree.

[ 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.

Two forces F₁ and F₂ act on a particle P. The force F₁ is given by F₁ = (–i + 2j) N and F₂ acts in the direction of the vector (i + j). Given that the resultant of F₁ and F₂ acts in the direction of the vector (i + 3j),

find F₂

The acceleration of P is (3i + 9j) m s^–2. At time t = 0, the velocity of P is (3i – 22j) m s^–1

Find the speed of P when t = 3 seconds.

Two particles P and Q have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle P is held at rest on a fixed rough horizontal table. The coefficient of friction between P and the table is . The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle Q hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle P is released from rest with the string taut and slides along the table. Assuming that P has not reached the pulley, find

the tension in the string during the motion,

the magnitude and direction of the resultant force exerted on the pulley by the string.

[In this question, the horizontal unit vectors i and j are directed due east and due north respectively.] The velocity, v m s^–1, of a particle P at time t seconds is given by v = (1 – 2 t) i + (3 t – 3) j

Find the speed of P when t = 0

Find the bearing on which P is moving when t = 2

Find the value of t when P is moving

parallel to j,

parallel to (– i – 3 j).

Two particles A and B have masses 2 m and 3 m respectively. The particles are attached to the ends of a light inextensible string. Particle A is held at rest on a smooth horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. Particle B hangs at rest vertically below the pulley with the string taut, as shown in Figure 2. Particle A is released from rest. Assuming that A has not reached the pulley, find

the acceleration of B,

the tension in the string,

the magnitude and direction of the force exerted on the pulley by the string.

[ 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.

A particle P is moving with constant velocity (3i 2j) m s¹. At time t 6 s P is at the point with position vector (4i 7j) m. Find the distance of P from the origin at time t 2 s.

[ 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.

A particle is acted upon by two forces F₁ and F₂, given by F₁ = (i – 3j) N, F₂ = (pi + 2pj) N, where p is a positive constant.

Find the angle between F₂ and j. The resultant of F₁ and F₂ is R. Given that R is parallel to i,

find the value of p.

[ 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.

A particle P of mass 0.4 kg moves under the action of a single constant force F newtons. The acceleration of P is (6i + 8j) m s^–2. Find

the angle between the acceleration and i,

the magnitude of F.

At time t seconds the velocity of P is v m s^–1. Given that when t = 0, v = 9i – 10j,

find the velocity of P when t = 5.

[ 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.