Scalars and vectors：練習問題

Vector \(A\) has a magnitude of \(10.0\text{ units}\) and acts at an angle of \(30.0^\circ\) north of east. Vector \(B\) has a magnitude of \(15.0\text{ units}\) and acts due west. What is the magnitude of the resultant vector \(R = A + B\)?

A force \(F\) of \(50.0\text{ N}\) acts at an angle of \(\theta\) to the horizontal. The horizontal component of the force is \(30.0\text{ N}\). A second force of \(20.0\text{ N}\) acts vertically upwards. What is the magnitude and direction (angle from horizontal) of the resultant of these two forces?

Three coplanar forces are in equilibrium. Force \(F_1\) is \(5.0\text{ N}\) acting due North. Force \(F_2\) is \(12.0\text{ N}\) acting due East. What is the magnitude and direction of the third force \(F_3\)?

A vector has a magnitude of \(V\). It is resolved into two perpendicular components. One component is \(V \sin(\theta)\). What is the magnitude of the other component in terms of the first component \(x\)?

A ship travels \(20\text{ km}\) on a bearing of \(060^\circ\) and then \(30\text{ km}\) on a bearing of \(150^\circ\). What is the magnitude of the total displacement of the ship from its starting point?

An aircraft is flying with a velocity of \(250\text{ m s}^{-1}\) at an angle of \(30^{\circ}\) North of East. Calculate the magnitude of the velocity component in the Northerly direction.

Three coplanar forces are in equilibrium. Two of the forces are \(8.0\text{ N}\) acting vertically upwards and \(6.0\text{ N}\) acting horizontally to the right. Determine the magnitude and the angle relative to the horizontal of the third force.

A boat travels at a velocity of \(5.0\text{ m s}^{-1}\) relative to the water, heading due North. The river has a current of \(2.0\text{ m s}^{-1}\) flowing due East. Calculate the magnitude and the bearing of the resultant velocity of the boat.

(a) Distinguish between scalar and vector quantities, giving one example of each from the study of kinematics.
(b) An aircraft is flying with a velocity of \( 120 \text{ m s}^{-1} \) relative to the air due North. A crosswind is blowing from the West to the East at a constant speed of \( 35 \text{ m s}^{-1} \).
(i) Use a vector triangle or calculation to determine the magnitude and direction of the resultant velocity of the aircraft relative to the ground.
(ii) If the pilot wishes to fly exactly North relative to the ground, calculate the angle at which the aircraft must be steered relative to the North.