Chapter Questions
Determine the magnitude of the resultant force $\mathbf{F}_R=\mathbf{F}_1+\mathrm{F}_3$ and its direction, measured counterclockwise from the positive $x$ axis.
Determine the magnitude of the resultant force if:(a) $\mathbf{F}_n=\mathbf{F}_1+\mathbf{F}_2 ;$ (b) $\mathbf{F}_R^Z=\mathbf{F}_1-\mathbf{F}_2$.
Determine the magnitude of the resultant force $\mathbf{F}_R=\mathrm{F}_1+\mathrm{F}_2$ and its direction, measured counterclockwise from the positive $x$ axis.
Determine the magnitude of the resultant force $\mathbf{F}_R=F_1+F_2$ and its direction, measured clockwise from the positive $u$ axis.
Resolve the foree $\mathbf{F}_1$ into components acting along the $u$ and $v$ axes and determine the magnitudes of the components.
Resolve the force $\mathrm{F}_2$ into components acting along the $u$ and $v$ axes and determine the magnitudes of the components.
The plate is subjected to the two forces at $A$ and $B$ as shown. If $\theta=60^{\circ}$, determine the magnitude of the resultant of these two forces and its direction measured from the horizontal.
Determine the angle $\theta$ for connecting member $A$ to the plate so that the resultant force of $F_A$ and $F_B$ is directed horizontally to the right. Also, what is the magnitude of the resultant force.
The vertical force $\mathbf{F}$ acts downward at $A$ on the twomembered frame. Determine the magnitudes of the two components of $\mathbf{F}$ directed along the axes of $A B$ and $A C$. Set $F=500 \mathrm{~N}$.
Solve Prob. $2-9$ with $F=350 \mathrm{lb}$.
The force acting on the gear tooth is $F=20 \mathrm{lb}$. Resolve this force into two components acting along the lines $a a$ and $b b$.
The component of force $\mathbf{F}$ acting along line $a a$ is require to be 30 lb . Determine the magnitude of $\mathbf{F}$ and its component along line $b b$.
The $500-1 \mathrm{~b}$ force acting on the frame is to be resolved into two components acting along the axis of the struts $A B$ and $A C$. If the component of force along $A C$ is required to be 300 lb , directed from $A$ to $C$, determine the magnitude of force acting along $A B$ and the angle $\theta$ of the $500-\mathrm{lb}$ force.
Determine the angle $\theta$ between the two cables attached to the post.
The post is to be pulled out of the ground using two ropes $A$ and $B$. Rope $A$ is subjected to a force of 600 lb and is directed at $60^{\circ}$ from the horizontal. If the resultant force acting on the post is to be 1200 lb , vertically upward, determine the force $T$ in rope $B$ and the corresponding angle $\theta$.
Determine the design angle $\theta\left(0^{\circ} \leq \theta \leq 90^{\circ}\right)$ for strut $A B$ so that the $400-\mathrm{lb}$ horizontal force has a component of $500-\mathrm{lb}$ directed from $A$ towards $C$. What is the component of force acting along member $A B$ ? Take $\phi=40^{\circ}$.
Determine the design angle $\phi\left(0^{\circ} \leqslant \phi \leqslant 90^{\circ}\right)$ between struts $A B$ and $A C$ so that the 400 -lb horizontal force has a component of $600-1 \mathrm{~b}$ which acts up to the left, in the same direction as from $B$ towards $A$. Take $\theta=30^{\circ}$.
The chisel exerts a force of 20 lb on the wood dowel rod which is turning in a lathe. Resolve this force into components acting (a) along the $n$ and $y$ axes and (b) along the $x$ and $t$ axes.
Two forces are applied at the end of a screw eye in order to remove the post. Determine the angle $\theta\left(0^{\circ} \leq\right.$ $\theta \leq 90^{\circ}$ ) and the magnitude of force $\mathbf{F}$ so that the resultant force acting on the post is directed vertically upward and has a magnitude of 750 N .
If $F_1=F_2=30 \mathrm{lb}$, determine the angles $\theta$ and $\phi$ so that the resultant force is directed along the positive $x$ axis and has a magnitude of $F_R=20 \mathrm{lb}$.
The truck is to be towed using two ropes. Determine the magnitude of forces $\mathbf{F}_A$ and $\mathbf{F}_B$ acting on each rope in order to develop a resultant force of 950 N directed along the positive $x$ axis. Set $\theta=50^{\circ}$.
The truck is to be towed using two ropes. If the resultant force is to be 950 N , directed along the positive $x$ axis, determine the magnitudes of forces $\mathbf{F}_A$ and $\mathbf{F}_B$ acting on each rope and the angle of $\theta$ of $\mathbf{F}_B$ so that the magnitude of $\mathbf{F}_B$ is a minimum. $\mathbf{F}_A$ acts at $20^{\circ}$ from the $x$ axis as shown.
Determine the magnitude and direction of the resultant $F_R=F_1+F_2+F_3$ of the three forces by first finding the resultant $\mathbf{F}^{\prime}=\mathbf{F}_1+\mathbf{F}_2$ and then forming $\mathbf{F}_R=\mathbf{F}^{\prime}+\mathbf{F}_3$.
Determine the magnitude and direction of the resultant $\mathbf{F}_R=\mathbf{F}_1+\mathbf{F}_2+\mathbf{F}_3$ of the three forces by first finding the resultant $\mathbf{F}^{\prime}=\mathbf{F}_2+\mathbf{F}_3$ and then forming $\mathbf{F}_R=\mathbf{F}^{\prime}+\mathbf{F}_1$.
Resolve the $50-1 b$ force into components acting along (a) the $x$ and $y$ axes, and (b) the $x$ and $y^{\prime}$ axes.
The $\log$ is being towed by two tractors $A$ and $B$. Determine the magnitude of the two towing forces $\mathbf{F}_{\mathrm{A}}$ and $\mathbf{F}_g$ if it is required that the resultant force have a magnitude $F_R=10 \mathrm{kN}$ and be directed along the $x$ axis. $\operatorname{Set} \theta=15^{\circ}$.
If the resultant $\mathbf{F}_R$ of the two forces acting on the $\log$ is to be directed along the positive $x$ axis and have a magnitude of 10 kN , determine the angle $\theta$ of the cable, attached to $B$ such that the force $\mathbf{F}_b$ in this cable is minimum. What is the magnitude of the force in each cable for this situation?
The beam is to be hoisted using two chains. Determine the magnitudes of forces $\mathbf{F}_A$ and $\mathbf{F}_B$ acting on each chain in order to develop a resultant force of 600 N directed along the positive $y$ axis. Set $\theta=45^{\circ}$.
The beam is to be hoisted using two chains. If the resultant force is to be 600 N , directed along the positive $y$ axis, determine the magnitudes of forces $\mathbf{F}_A$ and $\mathbf{F}_n$ acting on each chain and the orientation $\theta$ of $\hat{F}_B$ so that the magnitude of $\mathbf{F}_h$ is a minimum. $\mathbf{F}_A$ acts at $30^{\circ}$ from the $y$ axis as shown.
Three chains act on the bracket such that they create a resultant force having a magnitude of 500 lb . If two of the chains are subjected to known forces, as shown, determine the orientation $\theta$ of the third chain, measured clockwise from the positive $x$ axis, so that the magnitude of force $\mathbf{F}$ in this chain is a minimum. All forces lie in the $x-y$ plane. What is the magnitude of $\mathbf{F}$ ? Hint: First find the resultant of the two known forces. Force $\mathbf{F}$ acts in this direction.
Three cables pull on the pipe such that they create a resultant force having a magnitude of 900 lb . If two of the cables are subjected to known forces, as shown in the figure, determine the direction $\theta$ of the third cable so that the magnitude of force $\mathbf{F}$ in this cable is a minimum. All forces lie in the $x-y$ plane. What is the magnitude of $\mathbf{F}$ ? Hint: First find the resultant of the two known forces.
Determine the $x$ and $y$ components of the $800-\mathrm{lb}$ force.
Determine the magnitude of the resultant force and its direction, measured clockwise from the positive $x$ axis.
Determine the magnitude of force $\mathbf{F}$ so that the resultant $\mathbf{F}_R$ of the three forces is as small as possible.
. Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive $x$ axis.
Three forces act on the bracket. Determine the magnitude and direction $\theta$ of $\mathbf{F}_1$ so that the resultant force is directed along the positive $x^{\prime}$ axis and has a magnitude of 1 kN .
If $F_1=300 \mathrm{~N}$ and $\theta=20^{\circ}$, determine the magnitude and direction, measured counterclockwise from the $x^{\prime}$ axis, of the resultant force of the three forces acting on the bracket.
Determine the magnitude and direction $\theta$ of $\mathbf{F}_1$ so that the resultant force is directed vertically upward and has a magnitude of 800 N .
Determine the magnitude and direction measured counterclockwise from the positive $x$ axis of the resultant force of the three forces acting on the ring $A$. Take $F_1=$ 500 N and $\theta=20^{\circ}$.
Express $\mathbf{F}_1$ and $\mathbf{F}_2$ as Cartesian vectors.
Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive $x$ axis.
Solve Prob. 2-1 by summing the rectangular or $x, y$ components of the forces to obtain the resultant force.
Solve Prob. 2-22 by summing the rectangular or $x, y$ components of the forces to obtain the resultant force.
Determine the magnitude and orientation $\theta$ of $\mathbf{F}_B$ so that the resultant force is directed along the positive $y$ axis and has a magnitude of 1500 N .
Determine the magnitude and orientation, measured counterclockwise from the positive $y$ axis, of the resultant force acting on the bracket, if $F_H=600 \mathrm{~N}$ and $\theta=20^{\circ}$.
Determine the $x$ and $y$ components of $\mathbf{F}_1$ and $\mathbf{F}_2$.
Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive $x$ axis.
Determine the $x$ and $y$ components of each force acting on the gusset plate of the bridge truss. Show that the resultant force is zero.
If $\theta=60^{\circ}$ and $F=20 \mathrm{kN}$, determine the magnitude of the resultant force and its direction measured clockwise from the positive $x$ axis.
Determine the magnitude and direction $\theta$ of $\mathbf{F}_A$ so that the resultant force is directed along the positive $x$ axis and has a magnitude of 1250 N .
Determine the magnitude and direction, measured counterclockwise from the positive $x$ axis, of the resultant force acting on the ring at $O$, if $F_A=750 \mathrm{~N}$ and $\theta=45^{\circ}$.
Express each of the three forces acting on the column in Cartesian vector form and compute the magnitude of the resultant force.
The three concurrent forces acting on the screw eye produce a resultant force $\mathbf{F}_R=0$. If $F_2={ }_3^2 F_1$ and $\mathbf{F}_1$ is to be $90^{\circ}$ from $\mathbf{F}_2$ as shown, determine the required magnitude of $\mathbf{F}_3$ expressed in terms of $F_1$ and the angle $\theta$.
Determine the magnitude of force $\mathbf{F}$ so that the resultant $\mathbf{F}_R$ of the three forces is as small as possible. What is the minimum magnitude of $\mathbf{F}_R$ ?
. Express each of the three forces acting on the bracket in Cartesian vector form with respect to the $x$ and $y$ axes. Determine the magnitude and direction $\theta$ of $\mathrm{F}_1$ so that the resultant force is directed along the positive $x^*$ axis and has a magnitude of $F_R=600 \mathrm{~N}$.
The three concurrent forces acting on the post produce a resultant force $\mathbf{F}_R=0$. If $F_2=\frac{1}{2} F_1$, and $\mathbf{F}_1$ is to be $90^{\circ}$ from $F_2$ as shown, determine the required magnitude $F_3$ expressed in terms of $F_1$ and the angle $\theta$.
Three forces act on the bracket. Determine the magnitude and orientation $\theta$ of $\mathbf{F}_2$ so that the resultant force is directed along the positive $u$ axis and has a magnitude of 50 lb .
If $F_2=150 \mathrm{lb}$ and $\theta=55^{\circ}$, determine the magnitude and orientation, measured clockwise from the positive $x$ axis, of the resultant force of the three forces acting on the bracket.
Determine the magnitude of force $\mathbf{F}$ so that the resultant force of the three forces is as small as possible. What is the magnitude of the resultant force?
Determine the magnitude and coordinate direction angles of $\mathbf{F}_1=\{60 \mathrm{i}-50 \mathrm{j}+40 \mathbf{k}\} \mathrm{N}$ and $\mathbf{F}_2=\{-40 \mathrm{i}-$ $85 \mathbf{j}+30 \mathrm{k}$ / N. Sketch each force on an $x, y, z$ reference.
The cable at the end of the crane boom exerts a force of 250 lb on the boom as shown. Express $\mathbf{F}$ as a Cartesian vector.
Determine the magnitude and coordinate direction angles of the force $\mathbf{F}$ acting on the stake.
Determine the magnitude and the coordinate direction angles of the resultant force.
The stock $S$ mounted on the lathe is subjected to a force of 60 N , which is caused by the die $D$. Determine the coordinate direction angle $\beta$ and express the force as a Cartesian vector.
Determine the magnitude and coordinate direction angles of the resultant force and sketch this vector on the coordinate system.
Specify the coordinate direction angles of $\mathbf{F}_1$ and $\mathrm{F}_2$ and express each force as a Cartesian vector.
The mast is subjected to the three forces shown. Determine the coordinate direction angles $\alpha_1, \beta_1, \gamma_1$ of $\mathbf{F}_1$ so that the resultant force acting on the mast is $\mathbf{F}_R=\{350 \mathrm{H}\} \mathrm{N}$.
The mast is subjected to the three forces shown. Determine the coordinate direction angles $a_1, \beta_1, \gamma_1$ of $\mathbf{F}_1$ so that the resultant force acting on the mast is zero.
The cables attached to the screw eye are subjected to the three forces shown. Express each force in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force.
The beam is subjected to the two forces shown. Express each force in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force.
The two forces $\mathbf{F}_1$ and $\mathbf{F}_2$ acting at $A$ have a resultant force of $\mathbf{F}_R=(-100 \mathrm{k}\} \mathrm{lb}$. Determine the magnitude and coordinate direction angles of $\mathbf{F}_2$.
Determine the coordinate direction angles of the force $\mathbf{F}_1$ and indicate them on the figure.
The bracket is subjected to the two forces shown. Express each force in Cartesian vector form and then determine the resultant force $\mathbf{F}_R$. Find the magnitude and coordinate direction angles of the resultant force.
The pole is subjected to the force $\mathbf{F}$, which has components acting along the $x, y, z$ axes as shown. If the magnitude of $\mathbf{F}$ is 3 kN , and $\beta=30^{\circ}$ and $\gamma=75^{\circ}$, determine the magnitudes of its three components.
The pole is subjected to the force $\mathbf{F}$ which has components $F_x=1.5 \mathrm{kN}$ and $F_z=1.25 \mathrm{kN}$. If $\beta=75^{\circ}$, determine the magnitudes of $\mathbf{F}$ and $\mathbf{F}_y$
A force $\mathbf{F}$ is applied at the top of the tower at $A$. If it acts in the direction shown such that one of its components lying in the shaded $y-z$ plane has a magnitude of 80 lb , determine its magnitude $F$ and coordinate direction angles $\alpha, \beta, \gamma$.
Three forces act on the hook. If the resultant force $\mathbf{F}_R$ has a magnitude and direction as shown, determine the magnitude and the coordinate direction angles of force $\mathbf{F}_3$.
Determine the coordinate direction angles of $\mathbf{F}_1$ and $F_R$.
The bolt is subjected to the force $\mathbf{F}$, which has components acting along the $x, y, z$ axes as shown. If the magnitude of $\mathbf{F}$ is 80 N , and $\alpha=60^{\circ}$ and $\gamma=45^{\circ}$, determine the magnitudes of its components.
Two forces $\mathbf{F}_1$ and $\mathbf{F}_2$ act on the bolt. If the resultant force $\mathbf{F}_R$ has a magnitude of 50 lb and coordinate direction angles $\alpha=110^{\circ}$ and $\beta=80^{\circ}$, as shown, determine the magnitude of $\mathbf{F}_2$ and its coordinate direction angles.
If $\mathbf{r}_1=\{3 \mathbf{i}-4 \mathbf{j}+3 \mathbf{k}\} \mathrm{m}, \mathbf{r}_2=\{4 \mathbf{i}-5 \mathbf{k}\} \mathrm{m}, \mathbf{r}_1=$ $\{3 \mathbf{i}-2 \mathbf{j}+5 \mathbf{k}] \mathrm{m}$, determine the magnitude and direction of $\mathbf{r}=2 \mathbf{r}_1-\mathbf{r}_2+3 \mathbf{r}_3$.
Represent the position vector $\mathbf{r}$ acting from point $A(3 \mathrm{~m}, 5 \mathrm{~m}, 6 \mathrm{~m})$ to point $B(5 \mathrm{~m},-2 \mathrm{~m}, 1 \mathrm{~m})$ in Cartesian vector form. Determine its coordinate direction angles and find the distance between points $A$ and $B$.
A position vector extends from the origin to point $A(2 \mathrm{~m}, 3 \mathrm{~m}, 6 \mathrm{~m})$. Determine the angles $\alpha, \beta, \gamma$ which the tail of the vector makes with the $x, y, z$ axes, respectively.
Express the position vector $\mathbf{r}$ in Cartesian vector form; then determine its magnitude and coordinate direction angles.
Express force $\mathbf{F}$ as a Cartesian vector; then determine its coordinate direction angles,
Determine the length of member $A B$ of the truss by first establishing a Cartesian position vector from $A$ to $B$ and then determining its magnitude.
At a given instant, the position of a plane at $A$ and a train at $B$ are measured relative to a radar antenna at $O$. Determine the distance $d$ between $A$ and $B$ at this instant. To solve the problem, formulate a position vector, directed from $A$ to $B$, and then determine its magnitude.
The hinged plate is supported by the cord $A B$. If the force in the cord is $F=340 \mathrm{lb}$, express this force, directed from $A$ toward $B$, as a Cartesian vector. What is the length of the cord?
Determine the length of the crankshaft $A B$ by first formulating a Cartesian position vector from $A$ to $B$ and then determining its magnitude.
. Determine the lengths of wires $A D, B D$, and $C D$. The ring at $D$ is midway between $A$ and $B$.
Express force $\mathbf{F}$ as a Cartesian vector; then determine its coordinate direction angles.
Determine the magnitude and coordinate direction angles of the resultant force acting at point $A$.
The door is held opened by means of two chains. If the tension in $A B$ and $C D$ is $F_A=300 \mathrm{~N}$ and $F_C=250$ N , respectively, express each of these forces in Cartesian vector form.
The two mooring cables exert forces on the stern of a ship as shown. Represent each force as as Cartesian vector and determine the magnitude and direction of the resultant.
Two tractors pull on the tree with the forces shown. Represent each force as a Cartesian vector and then determine the magnitude and coordinate direction angles of the resultant force.
The guy wires are used to support the telephone pole. Represent the force in each wire in Cartesian vector form.
Express each of the forces in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force.
The cable attached to the tractor at $B$ exerts a force of 350 lb on the framework. Express this force as a Cartesian vector.
The load at $A$ creates a force of 60 lb in wire $A B$. Express this force as a Cartesian vector acting on $A$ and directed toward $B$ as shown.
The pipe is supported at its ends by a cord $A B$. If the cord exerts a force of $F=12 \mathrm{lb}$ on the pipe at $A$, express this force as a Cartesian vector.
The cord exerts a force of $\mathbf{F}=\{12 \mathbf{i}+9 \mathbf{j}-8 \mathbf{k}\} \mathrm{lb}$ on the hook. If the cord is 8 ft long, determine the location $x, y$ of the point of attachment $B$, and the height $z$ of the hook.
The cord exerts a force of $F=30 \mathrm{lb}$ on the hook. If the cord is 8 ft long, $z=4 \mathrm{ft}$, and the $x$ component of the force is $F_x=25 \mathrm{lb}$, determine the location $x, y$ of the point of attachment $B$ of the cord to the ground.
Each of the four forces acting at $E$ has a magnitude of 28 kN . Express each force as a Cartesian vector and determine the resultant force.
The tower is held in place by three cables If the force of each cable acting on the tower is shown, determine the magnitude and coordinate direction angles $\alpha, \beta, \gamma$ of the resultant force. Take $x=20 \mathrm{~m}, y=15 \mathrm{~m}$.
The cable, attached to the shear-leg derrick, exerts a force on the derrick of $F=350 \mathrm{lb}$. Express this force as a Cartesian vector.
The window is held open by chain $A B$. Determine the length of the chain, and express the $50-\mathrm{lb}$ force acting at $A$ along the chain as a Cartesian vector and determine its coordinate direction angles.
Given the three vectors A, B, and D, show that
$$\mathbf{A} \cdot(\mathbf{B}+\mathbf{D})=(\mathbf{A} \cdot \mathbf{B})+(\mathbf{A} \cdot \mathbf{D})$$
Determine the angle $\theta$ between the tails of the two vectors.
Determine the magnitude of the projected component of $\mathbf{r}_1$ along $\mathbf{r}_2$, and the projection of $\mathbf{r}_2$ along $\mathbf{r}_1$.
Determine the angle $\theta$ between the $y$ axis of the pole and the wire $A B$.
The force $\mathbf{F}=\{25 \mathbf{i}-50 \mathbf{j}+10 \mathbf{k}] \mathrm{N}$ acts at the end $A$ of the pipe assembly. Determine the magnitude of the components $\mathbf{F}_1$ and $\mathbf{F}_2$ which act along the axis of $A B$ and perpendicular to it.
Determine the angle $\theta$ between the sides of the triangular plate.
Determine the length of side $B C$ of the triangular plate. Solve the problem by finding the magnitude of $\mathbf{r}_{H C}$; then check the result by first finding $\theta, r_{A B}$, and $r_{A C}$ and then use the cosine law.
Determine the components of $\mathbf{F}$ that act along rod $A C$ and perpendicular to it. Point $B$ is located at the midpoint of the rod.
Determine the components of $\mathbf{F}$ that act along rod $A C$ and perpendicular to it. Point $B$ is located 3 m along the rod from end $C$.
The clamp is used on a jig. If the vertical force acting on the bolt is $\mathbf{F}=\{-500 \mathrm{k}\} \mathrm{N}$, determine the magnitudes of the components $\mathbf{F}_1$ and $\mathbf{F}_2$ which act along the $O A$ axis and perpendicular to it.
Determine the projection of the force $\mathbf{F}$ along the pole.
Determine the projected component of the $80-\mathrm{N}$ force acting along the axis $A B$ of the pipe.
Cable $O A$ is used to support column $O B$. Determine the angle $\theta$ it makes with beam $O C$.
Cable $O A$ is used to support column $O B$. Determine the angle $\phi$ it makes with beam $O D$.
The force $F$ acts at the end $A$ of the pipe assembly. Determine the magnitudes of the components $\mathbf{F}_1$ and $\mathbf{F}_2$ which act along the axis of $A B$ and perpendicular to it.
Two cables exert forces on the pipe. Determine the magnitude of the projected component of $\mathbf{F}_1$ along the line of action of $\mathbf{F}_2$.
Determine the angle $\theta$ between the two cables attached to the pipe.
Determine the angle $\theta$ between cables $A B$ and $A C$.
If $\mathbf{F}$ has a magnitude of 55 Ib , determine the magnitude of its projected component acting along the $x$ axis and along cable $A C$.
Determine the angle $\theta$ between the edges of the sheet-metal bracket.
The cables each exert a force of 400 N on the post. Determine the magnitude of the projected component of $\mathbf{F}_1$ along the line of action of $\mathbf{F}_2$.
Determine the angles $\theta$ and $\phi$ made between the axes $O A$ of the flag pole and $A B$ and $A C$, respectively, of each cable.
Determine the magnitude and coordinate direction angles of $\mathbf{F}_3$ so that the resultant of the three forces acts along the positive $y$ axis and has a magnitude of 600 lb .
Determine the magnitude and coordinate direction angles of $F_3$ so that the resultant of the three forces is zero.
Determine the design angle $\theta\left(\theta<90^{\circ}\right)$ between the two struts so that the $500-\mathrm{lb}$ horizontal force has a component of $600-\mathrm{lb}$ directed from $A$ toward $C$. What is the component of force acting along member $B A$ ?
The force $\mathbf{F}$ has a magnitude of 80 lb and acts at the midpoint $C$ of the thin rod. Express the force as a Cartesian vector.
Two forces $\mathbf{F}_1$ and $\mathbf{F}_2$ act on the hook. If their lines of action are at an angle $\theta$ apart and the magnitude of each force is $F_1=F_2=F$, determine the magnitude of the resultant force $\mathbf{F}_R$ and the angle between $\mathbf{F}_R$ and $\mathbf{F}_1$.
Determine the angles $\theta$ and $\phi$ between the wire segments.
Determine the magnitudes of the projected components of the force $\mathbf{F}=\{60 \mathrm{i}+12 \mathrm{j}-40 \mathrm{k}\} \mathrm{N}$ in the direction of the cables $A B$ and $A C$.
Determine the magnitude of the projected component of the $100-\mathrm{lb}$ force acting along the axis $B C$ of the pipe.
$$\begin{aligned}& \mathrm{w}_{c D}=\frac{(0-6) i+(12-4)]+[0-(-2)] \mathbf{k}}{\sqrt{(0-6)^2+(12-4)^2+[0-(-2)]^2}} \\& =-0.5883 i+0.7845 j+0.1961 k \\& \mathrm{~F}=F \mathrm{u}_{c D}=100(-0.5883 \mathrm{i}+0.7845 \mathrm{j}+0.1961 \mathrm{k}) \\& =\{-58.835 i+78.446 j+19.612 k\} \mathrm{db}\end{aligned}$$
The boat is to be pulled onto the shore using two ropes. If the resultant force is to be 80 lb , directed along the keel aa, as shown, determine the magnitudes of forces $\mathbf{T}$ and $\mathbf{P}$ acting in each rope and the angle $\theta$ of $\mathbf{P}$ is a minimum, $\mathbf{T}$ acts at $30^{\circ}$ from the keel as shown.