Vectors and resultantsEdexcel International A Level Maths: Subtopic test

1. 1
   Two forces act on a particle: $\mathbf{F}_1=(7\mathbf{i}-3\mathbf{j})$ N and $\mathbf{F}_2=(-2\mathbf{i}+6\mathbf{j})$ N, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in the horizontal plane, in the directions east and north.

   (a)

   Find the resultant of $\mathbf{F}_1$ and $\mathbf{F}_2$.

   [1 mark]

   • A$(9\mathbf{i}-9\mathbf{j})$ N
   • B$(-5\mathbf{i}-3\mathbf{j})$ N
   • C$(5\mathbf{i}+3\mathbf{j})$ N
   • D$(5\mathbf{i}-9\mathbf{j})$ N

   (b)

   What is the magnitude of the resultant force?

   [1 mark]

   • A5.83 N
   • B34 N
   • C8 N
   • D4 N

   (c)

   Find the angle between the resultant force and the direction east, giving your answer in degrees to 1 decimal place.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A force $\mathbf{F}$ of magnitude 24 N acts on a particle in a direction $40^\circ$ above the direction of the unit vector $\mathbf{i}$, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a vertical plane, horizontal and vertically upwards respectively.

   (a)

   Which expression gives the component of $\mathbf{F}$ in the direction of $\mathbf{i}$?

   [1 mark]

   • A$24\sin40^\circ$
   • B$24\tan40^\circ$
   • C$\dfrac{24}{\cos40^\circ}$
   • D$24\cos40^\circ$

   (b)

   What is the component of $\mathbf{F}$ in the direction of $\mathbf{j}$?

   [1 mark]

   • A18.4 N
   • B20.1 N
   • C15.4 N
   • D31.3 N

   (c)

   A second force $\mathbf{G}=(-10\mathbf{i}-6\mathbf{j})$ N also acts on the particle. Find the magnitude of the resultant of $\mathbf{F}$ and $\mathbf{G}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three forces $\mathbf{P}=(3\mathbf{i}+5\mathbf{j})$ N, $\mathbf{Q}=(-7\mathbf{i}+2\mathbf{j})$ N and $\mathbf{R}=(p\mathbf{i}+q\mathbf{j})$ N act on a particle, where $p$ and $q$ are constants. The particle is in equilibrium.

   (a)

   Find the values of $p$ and $q$.

   [3 marks]

   (b)

   Find the magnitude of $\mathbf{R}$, and the angle $\mathbf{R}$ makes with the direction of $\mathbf{i}$, stating whether it is above or below $\mathbf{i}$. Give your answers to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle is acted on by two forces. The force $\mathbf{F}_1$ has magnitude 26 N and acts in the direction of the vector $5\mathbf{i}+12\mathbf{j}$. The force $\mathbf{F}_2=(-4\mathbf{i}+7\mathbf{j})$ N. The vectors $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a horizontal plane.

   (a)

   Express $\mathbf{F}_1$ in terms of $\mathbf{i}$ and $\mathbf{j}$. Hence find the magnitude of the resultant of $\mathbf{F}_1$ and $\mathbf{F}_2$.

   [6 marks]

   (b)

   A third force $\mathbf{F}_3$ is added so that the resultant of the three forces is 20 N in the direction of $\mathbf{i}$. Find $\mathbf{F}_3$, its magnitude, and the acute angle between $\mathbf{F}_3$ and the direction of $-\mathbf{j}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions