Answer: Given the three vectors A, B, and D, show that A #

If \(\theta = 60^{\circ}\) and F = 450 N, determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

If the magnitude of the resultant force is to be 500 N, directed along the positive y axis, determine the magnitude of force F and its direction \(\theta\).

Determine the magnitude of the resultant force \(F_R = F_1 + F_2\) and its direction, measured counterclockwise from the positive x axis.

The vertical force F acts downward at A on the twomembered frame. Determine the magnitudes of the two components of F directed along the axes of AB and AC. Set F = 500 N.

Solve Prob. 2–4 with F = 350 lb.

Determine the magnitude of the resultant force \(F_R = F_1 + F_2\) and its direction, measured clockwise from the positive u axis.

Resolve the force \(F_1\) into components acting along the u and v axes and determine the magnitudes of the components.

Resolve the force \(F_2\) into components acting along the u and v axes and determine the magnitudes of the components.

If the resultant force acting on the support is to be 1200 lb, directed horizontally to the right, determine the force F in rope A and the corresponding angle \(\theta\).

Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

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 clockwise from the horizontal.

Determine the angle u 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 force acting on the gear tooth is F = 20 lb. Resolve this force into two components acting along the lines aa and bb.

The component of force F acting along line aa is required to be 30 lb. Determine the magnitude of F and its component along line bb.

Force F acts on the frame such that its component acting along member AB is 650 lb, directed from B towards A, and the component acting along member BC is 500 lb, directed from B towards C. Determine the magnitude of F and its direction \(\theta\). Set \(\phi = 60^{\circ}\).

Force F acts on the frame such that its component acting along member AB is 650 lb, directed from B towards A. Determine the required angle \(\phi (0^{\circ}\ \leq\ \phi\ \leq\ 45^{\circ})\) and the component acting along member BC. Set F = 850 lb and \(\theta = 30^{\circ}\).

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 \(F = F_1 + F_2\) and then forming \(F_R = F’ + F_3\).

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 \(F’ = F_2 + F_3\) and then forming \(F_R = F’ + F_1\).

Determine the design angle \(\theta (0^{\circ}\ \leq\ \theta\ \leq\ 90^{\circ})\) for strut AB so that the 400-lb horizontal force has a component of 500 lb directed from A towards C. What is the component of force acting along member AB? Take \(\phi = 40^{\circ}\).

Determine the design angle \(\phi (0^{\circ}\ \leq\ \phi\ \leq\ 90^{\circ})\) between struts AB and AC so that the 400-lb horizontal force has a component of 600 lb which acts up to the left, in the same direction as from B towards A. Take \(\theta = 30^{\circ}\).

Determine the magnitude and direction of the resultant force, \(F_R\) measured counterclockwise from the positive x axis. Solve the problem by first finding the resultant \(F’ = F_1 + F_2\) and then forming \(F_R = F’ + F_3\).

Determine the magnitude and direction of the resultant force, measured counterclockwise from the positive x axis. Solve l by first finding the resultant \(F’ = F_2 + F_3\) and then forming \(F_R = F’ + F_1\).

Two forces act on the screw eye. If \(F_1 = 400\ N\) and \(F_2 = 600\ N\0, determine the angle \(\theta (0^{\circ}\ \leq\ \theta\ \leq\ 180^{\circ})\) between them, so that the resultant force has a magnitude of \(F_R = 800\ N\).

Two forces \(F_1\) and \(F_2\) act on the screw eye. 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 \(F_R\) and the angle between \(F_R\) and \(F_1\).

If \(F_1 = 30\ lb\) and \(F_2 = 40\ lb\), determine the angles \(\theta\) and \(\phi\) so that the resultant force is directed along the positive x axis and has a magnitude of \(F_R = 60\ lb\).

Determine the magnitude and direction \(\theta\) of \(F_A\) 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 \(F_A = 750\ N\) and \(\theta = 45^{\circ}\).

Determine the magnitude of force F so that the resultant \(F_R\) of the three forces is as small as possible. What is the minimum magnitude of \(F_R\)?

If the resultant force of the two tugboats is 3 kN, directed along the positive x axis, determine the required magnitude of force \(F_B\) and its direction \(\theta\).

If \(F_B = 3\ kN\) and \(\theta = 45^{\circ}\), determine the magnitude of the resultant force of the two tugboats and its direction measured clockwise from the positive x axis.

If the resultant force of the two tugboats is required to be directed towards the positive x axis, and \(F_B\) is to be a minimum, determine the magnitude of \(F_R\) and \(F_B\) and the angle \(\theta\).

Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

Determine the magnitude of the resultant force and its direction, measured clockwise from the positive x axis.

Resolve \(F_1\) and \(F_2\) into their x and y components.

Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.

Resolve each force acting on the gusset plate into its x and y components, and express each force as a Cartesian vector.

Determine the magnitude of the resultant force acting on the plate and its direction, measured counterclockwise from the positive x axis.

Express each of the three forces acting on the support in Cartesian vector form and determine the magnitude of the resultant force and its direction, measured clockwise from positive x axis.

Determine the x and y components of \(F_1\) and \(F_2\).

Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

Express \(F_1\), \(F_2\), and \(F_3\) as Cartesian vectors.

Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

Determine the magnitude of the resultant force and its direction, measured clockwise from the positive x axis.

Determine the magnitude and direction \(\theta\) of the resultant force \(F_R\). Express the result in terms of the magnitudes of the components \(F_1\) and \(F_2\) and the angle \(\phi\).

Determine the magnitude and orientation \(\theta\) of \(F_B\) 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 \(F_B = 600\ N\) and \(\theta = 20^{\circ}\).

Three forces act on the bracket. Determine the magnitude and direction \(\theta\) of \(F_1\) so that the resultant force is directed along the positive x’ axis and has a magnitude of 800 N.

If \(F_1 = 300\ N\) and \(\theta = 10^{\circ}\), determine the magnitude and direction, measured counterclockwise from the positive x’ axis, of the resultant force acting on the bracket.

Express \(F_1\), \(F_2\), and \(F_3\) as Cartesian vectors.

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 a bridge truss. Show that the resultant force is zero.

Express \(F_1\) and \(F_2\) as Cartesian vectors.

Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.

Determine the magnitude of force F so that the resultant force of the three forces is as small as possible. What is the magnitude of the resultant force?

If the magnitude of the resultant force acting on the bracket is to be 450 N directed along the positive u axis, determine the magnitude of \(F_1\) and its direction \(\phi\).

If the resultant force acting on the bracket is required to be a minimum, determine the magnitudes of \(F_1\) and the resultant force. Set \(\phi = 30^{\circ}\).

Three forces act on the bracket. Determine the magnitude and direction \(\theta\) of F so that the resultant force is directed along the positive x’ axis and has a magnitude of 8 kN.

If F = 5 kN and \(\theta = 30^{\circ}\), determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

The force F has a magnitude of 80 lb and acts within the octant shown. Determine the magnitudes of the x, y, z components of F.

The bolt is subjected to the force F, which has components acting along the x, y, z axes as shown. If the magnitude of F is 80 N, and \(\alpha = 60^{\circ}\) and \(\gamma = 45^{\circ}\), determine the magnitudes of its components.

Determine the magnitude and coordinate direction angles of the force F acting on the support. The component of F in the x–y plane is 7 kN.

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 \(F_1\) and \(F_2\) and express each force as a Cartesian vector.

The screw eye is subjected to the two forces shown. Express each force in Cartesian vector form and then determine the resultant force. Find the magnitude and coordinate direction angles of the resultant force.

Determine the coordinate direction angles of \(F_1\).

Determine the magnitude and coordinate direction angles of \(F_3\) 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 magnitude and coordinate direction angles of the resultant force, and sketch this vector on the coordinate system.

Determine the magnitude and coordinate direction angles of the resultant force, and sketch this vector on the coordinate system.

Specify the magnitude and coordinate direction angles \(\alpha_1\), \(beta_1\), \(gamma_1\) of \(F_1\) so that the resultant of the three forces acting on the bracket is \(FR = \{-350k\}\ lb\). Note that \(F_3\) lies in the x–y plane.

Two forces \(F_1\) and \(F_2\) act on the screw eye. If the resultant force \(F_R\) has a magnitude of 150 lb and the coordinate direction angles shown, determine the magnitude of \(F_2\) and its coordinate direction angles.

Express each force in Cartesian vector form.

Determine the magnitude and coordinate direction angles of the resultant force, and sketch this vector on the coordinate system.

The spur gear is subjected to the two forces caused by contact with other gears. Express each force as a Cartesian vector.

The spur gear is subjected to the two forces caused by contact with other gears. Determine the resultant of the two forces and express the result as a Cartesian vector.

Determine the magnitude and coordinate direction angles of the resultant force, and sketch this vector on the coordinate system.

The two forces \(F_1\) and \(F_2\) acting at A have a resultant force of \(F_R = \{-100k\}\ lb\). Determine the magnitude and coordinate direction angles of \(F_2\).

Determine the coordinate direction angles of the force \(F_1\) 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 \(F_R\). Find the magnitude and coordinate direction angles of the resultant force.

If the coordinate direction angles for \(F_3\) are \(\alapha_3 = 120^{\circ}\), \(\beta_3 = 60^{\circ}\) and \(\gamma_3 = 45^{\circ}\), determine the magnitude and coordinate direction angles of the resultant force acting on the eyebolt.

If the coordinate direction angles for \(F_3\) are \(\alpha_3 = 120^{\circ}\), \(\beta_3 = 45^{\circ}\), and \(\gamma_3 = 60^{\circ}\), determine the magnitude and coordinate direction angles of the resultant force acting on the eyebolt.

If the direction of the resultant force acting on the eyebolt is defined by the unit vector \(u_{F_R} = cos\ 30^{\circ} j +sin\ 30^{\circ} k\), determine the coordinate direction angles of \(F_3\) and the magnitude of \(F_R\).

The pole is subjected to the force F, which has components acting along the x, y, z axes as shown. If the magnitude of F is 3 kN, \(\beta = 30^{\circ}\), and \(\gamma = 75^{\circ}\), determine the magnitudes of its three components.

The pole is subjected to the force F which has components \(F_x = 1.5\ kN\) and \(F_z = 1.25\ kN\). If \(\beta = 75^{\circ}\), determine the magnitudes of F and \(F_y\).

Determine the length of the connecting rod AB by first formulating a Cartesian position vector from A to B and then determining its magnitude.

Express force F as a Cartesian vector; then determine its coordinate direction angles.

Express each of the forces in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force.

If \(F = \{350i - 250j - 450k\}\ N\) and cable AB is 9 m long, determine the x, y, z coordinates of point A.

The 8-m-long cable is anchored to the ground at A. If x = 4 m and y = 2 m, determine the coordinate z to the highest point of attachment along the column.

The 8-m-long cable is anchored to the ground at A. If z = 5 m, determine the location +x, +y of point A. Choose a value such that x = y.

Express each of the forces in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force.

If \(F_B = 560\ N\) and \(F_C = 700\ N\), determine the magnitude and coordinate direction angles of the resultant force acting on the flag pole.

If \(F_B = 700\ N\), and \(F_C = 560\ N\), determine the magnitude and coordinate direction angles of the resultant force acting on the flag pole.

The plate is suspended using the three cables which exert the forces shown. Express each force as a Cartesian vector.

The three supporting cables exert the forces shown on the sign. Represent each force as a Cartesian vector.

Determine the magnitude and coordinate direction angles of the resultant force of the two forces acting on the sign at point A.

The force F has a magnitude of 80 lb and acts at the midpoint C of the thin rod. Express the force as a Cartesian vector.

The load at A creates a force of 60 lb in wire AB. Express this force as a Cartesian vector acting on A and directed toward B as shown.

Determine the magnitude and coordinate direction angles of the resultant force acting at point A on the post.

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 coordinate direction angles of the resultant.

The engine of the lightweight plane is supported by struts that are connected to the space truss that makes up the structure of the plane. The anticipated loading in two of the struts is shown. Express each of those forces as Cartesian vector.

Determine the magnitude and coordinate direction angles of the resultant force.

If the force in each cable tied to the bin is 70 lb, determine the magnitude and coordinate direction angles of the resultant force.

If the resultant of the four forces is \(F_R = \{-360k\}\ lb\), determine the tension developed in each cable. Due to symmetry, the tension in the four cables is the same.

Express the force F in Cartesian vector form if it acts at the midpoint B of the rod.

Express force F in Cartesian vector form if point B is located 3 m along the rod from end C.

The chandelier is supported by three chains which are concurrent at point O. If the force in each chain has a magnitude of 60 lb, express each force as a Cartesian vector and determine the magnitude and coordinate direction angles of the resultant force.

The chandelier is supported by three chains which are concurrent at point O. If the resultant force at O has a magnitude of 130 lb and is directed along the negative z axis, determine the force in each chain.

The window is held open by chain AB. Determine the length of the chain, and express the 50-lb force acting at A along the chain as a Cartesian vector and determine its coordinate direction angles.

The window is held open by cable AB. Determine the length of the cable and express the 30-N force acting at A along the cable as a Cartesian vector.

Given the three vectors A, B, and D, show that \(A \cdot (B + D) = (A \cdot B) + (A \cdot D)\).

Determine the magnitudes of the components of F = 600 N acting along and perpendicular to segment DE of the pipe assembly.

Determine the angle u between the two cables.

Determine the magnitude of the projection of the force \(F_1\) along cable AC.

Determine the angle u between the y axis of the pole and the wire AB.

Determine the magnitudes of the projected components of the force \(F = [60i + 12j - 40k]\ N\) along the cables AB and AC.

Determine the angle \(\theta\) between cables AB and AC.

A force of \(F = \{-40k\}\ lb\) acts at the end of the pipe. Determine the magnitudes of the components \(F_1\) and \(F_2\) which are directed along the pipe’s axis and perpendicular to it.

Two cables exert forces on the pipe. Determine the magnitude of the projected component of \(F_1\) along the line of action of \(F_2\).

Determine the angle \(\theta\) between the two cables attached to the pipe.

Determine the angle \(\theta\) between the cables AB and AC.

Determine the magnitude of the projected component of the force \(F = \{400i - 200j + 500k\}\ N\) acting along the cable BA.

Determine the magnitude of the projected component of the force \(F = \{400i - 200j + 500k\}\ N\) acting along the cable CA.

Determine the magnitude of the projection of force F = 600 N along the u axis.

Determine the magnitude of the projected component of the 100-lb force acting along the axis BC of the pipe.

Determine the angle \(\theta\) between pipe segments BA and BC.

Determine the angle \(\theta\) between BA and BC.

Determine the magnitude of the projected component of the 3 kN force acting along the axis BC of the pipe.

Determine the angles \(\theta\) and \(\phi\) made between the axes OA of the flag pole and AB and AC, respectively, of each cable.

Determine the magnitudes of the components of F acting along and perpendicular to segment BC of the pipe assembly.

Determine the magnitude of the projected component of F along AC. Express this component as a Cartesian vector.

Determine the angle \(\theta\) between the pipe segments BA and BC.

If the force F = 100 N lies in the plane DBEC, which is parallel to the x–z plane, and makes an angle of \(10^{\circ}\) with the extended line DB as shown, determine the angle that F makes with the diagonal AB of the crate.

Determine the magnitudes of the components of the force F = 90 lb acting parallel and perpendicular to diagonal AB of the crate.

Determine the magnitudes of the projected components of the force F = 300 N acting along the x and y axes.

Determine the magnitude of the projected component of the force F = 300 N acting along line OA.

Determine the angle u between the two cables.

Determine the projected component of the force F = 12 lb acting in the direction of cable AC. Express the result as a Cartesian vector.