Magnitude and direction of a vector

The position vector of the point A is 0.5 i - 0.5 j.

Determine whether 0.5 i - 0.5 j is a unit vector.

Find the direction of 0.5 i - 0.5 j, giving your answer in degrees.

The position vector of the point B is 3.5i + cj, where c is a constant.

Given that = 5, determine the possible values of c.

In this question you should take the acceleration due to gravity to be 10 m s^-2.

The unit vectors i and j are in a vertical plane with i horizontal and j vertically upwards.

A small ball P is projected from a point O on horizontal ground into the air. When P is in the air, it is to be modelled as a particle moving under the influence of gravity only.

At time t = 2 seconds, P has velocity (2i - 8j) m s^-1. The position vector of a point on the trajectory of P is (x i + y j) m relative to O.

Show that y = 6x - 1.25x².

Determine the maximum height of P above the ground during its motion.

Determine the following as P passes through the point on its trajectory where x = 2.5.

• the speed of P

• the direction of motion of P

In reality P does not move only under the influence of gravity but is also subject to air resistance.

Explain how this would affect your answer to part (b).

A particle P is moving with constant acceleration (-5i + 2j)m s^-2. At time t = 0 seconds, P is at the origin and has velocity (i + 3j)m s^-1.

Find, in terms of i and j, the displacement of P at time t = 2 seconds.

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

Find, in terms of k, the length of OP.

Determine the value of k.

In this question the unit vectors i and j are in the directions east and north respectively.

A particle P is moving on a smooth horizontal surface under the action of a single force F N. At time t seconds, where t ⩾ 0, the velocity v ms^-1 of P, relative to a fixed origin O, is given by

v = (1 - 2t)i + (2t² + t - 13)j.

Show that P is never stationary.

Find, in terms of i and j, the acceleration of P at time t.

The mass of P is 0.5 kg.

Determine the magnitude of F when P is moving in the direction of the vector - 2 i + j. Give

your answer correct to 3 significant figures.

When t = 1, P is at the point with position vector .

Determine the bearing of P from O at time t = 1.5.

A particle P moves with constant acceleration (-4i + 2j)ms^-2. At time t = 0 seconds, P is moving with velocity (7i + 6j)ms^-1.

Determine the speed of P when t = 3.

Determine the change in displacement of P between t = 0 and t = 3.

The point A is such that the magnitude of is 8 and the direction of is 240°.

Show the point A on the axes provided in the Printed Answer Booklet.

Find the position vector of point A.

Give your answer in terms of i and j.

The point B has position vector 6 i.

Find the exact area of triangle AOB.

The point C is such that OABC is a parallelogram.

Find the position vector of C.

Give your answer in terms of i and j.