Section 1
Vector representation
[mechanics]$$\text { Fig. } 22$$Figure 22 shows two forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ acting on an object $\mathrm{O}$. Determine the resultant force, $\mathbf{R}=\mathbf{F}_{1}+\mathbf{F}_{2}$.
[mechanics]Determine the horizontal, $\mathbf{F}_{x}$, and the vertical, $\mathbf{F}_{y}$, components of the force $\mathbf{F}$ shown in Fig. $23 .$
?? [mechanics] A force of $12000 \mathrm{~N}$ is applied to a body at an angle of $45^{\circ}$ to the horizontal. Determine the vertical and horizontal components of the force.
[mechanics]Fig. 24Figure 24 shows a force, $\mathbf{F}$, of $10 \mathrm{kN}$ applied at an angle $\theta$. Obtain the horizontal and vertical components of the force.
Fig. 25In the triangle OAB shown in Fig. 25 , the point $\mathrm{M}$ is the midpoint of $\mathrm{OB}$. Determine AM in terms of a and $\mathbf{b}$.
Fig. 26Figure 26 shows a triangle $\mathrm{ABC}$ where $\mathrm{M}$ is the midpoint of $\mathrm{AC}$ and $\mathrm{N}$ is the midpoint of BC. Show that $\overrightarrow{M N}=\frac{1}{2} \overrightarrow{A B}$
Figure 27 shows forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ exerted on a particle O. The resultant force, $\mathbf{F}=\mathbf{F}_{1}+\mathbf{F}_{2}$, is horizontal. Determine $\mathbf{F}$ if $\mathbf{F}_{2}$ has a magnitude of $10 \mathrm{~N}$.Fig. 27
Figure 28 shows forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ applied to an object $\mathrm{O}$. The resultant force $\mathbf{R}$ is vertical.Fig. 28Determine the magnitude of $\mathbf{R}$ if $\mathbf{F}_{1}$ has a magnitude of $12 \mathrm{kN}$.
Figure 29 shows forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ of magnitude $7 \mathrm{kN}$ and $20 \mathrm{kN}$ respectively. Determine the resultant force.Fig. 29
Determine the resultant force, $\mathbf{F}_{1}+\mathbf{F}_{2}$ of the forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ shown in Fig. 30Fig. 30