Q: Given a force $$\vec{F}$$ with magnitude $$F$$ and direction cosines $$\cos \theta_x$$, $$\cos \theta_y$$, and $$\cos \theta_z$$, which of the following must always be true?

$$\cos^2$ \theta_x + \cos^2$ \theta_y + \cos^2$ \theta_z = 1$$

Q: A cable exerts a force of $$2500\ \mathrm{N}$$ from point $$M(0,0,0) to $$point $$N(-40,80,30). $$What is the $$x-$$component of the force acting on the bolt at $$M$$?

$$F_x = 2500 \times \dfrac{-40}{94.34} = -1060\ \mathrm{N}$$

Q: Which of the following statements about the vector (cross) product is correct?

The vector product is not commutative: $$\vec{A} \times \vec{B} \neq \vec{B} \times \vec{A}$$

Q: If $$\vec{A} = 2\vec{i} + 3\vec{j} + 4\vec{k}$$ and $$\vec{B} = \vec{i} - 2\vec{j} + 2\vec{k}$$, what is $$\vec{A} \cdot \vec{B}$$?

$$2 \times 1 + 3 \times (-2) + 4 \times 2 = 2 - 6 + 8 = 4$$

Q: A force $$\vec{F}$$ has direction cosines $$\cos \theta_x = -0.424$$, $$\cos \theta_y = 0.849$$, $$\cos \theta_z = 0.318. $$What is the angle $$\theta_x$$ between $$\vec{F}$$ and the $$x-$$axis?

$$\theta_x = \cos$$^{-1}$$(-0.424) = 115.09^\circ$$

Question 1

A force $$\vec{F} in $$space has components $$F_x = 20\ \mathrm{N}$$, $$F_y = -30\ \mathrm{N}$$, and $$F_z = 60\ \mathrm{N}. $$What is the magnitude of $$\vec{F}$$?

Accepted Answer

$$\sqrt{20^2$ + (-30)^2$ + 60^2$} = 70\ \mathrm{N}$$

Question 2

Given a force $$\vec{F}$$ with magnitude $$F$$ and direction cosines $$\cos 	heta_x$$, $$\cos 	heta_y$$, and $$\cos 	heta_z$$, which of the following must always be true?

Accepted Answer

$$\cos^2$ 	heta_x + \cos^2$ 	heta_y + \cos^2$ 	heta_z = 1$$

Question 3

A cable exerts a force of $$2500\ \mathrm{N}$$ from point $$M(0,0,0) to $$point $$N(-40,80,30). $$What is the $$x-$$component of the force acting on the bolt at $$M$$?

Accepted Answer

$$F_x = 2500 	imes \dfrac{-40}{94.34} = -1060\ \mathrm{N}$$

Question 4

Which of the following statements about the vector (cross) product is correct?

Accepted Answer

The vector product is not commutative: $$\vec{A} 	imes \vec{B} 
eq \vec{B} 	imes \vec{A}$$

Question 5

If $$\vec{A} = 2\vec{i} + 3\vec{j} + 4\vec{k}$$ and $$\vec{B} = \vec{i} - 2\vec{j} + 2\vec{k}$$, what is $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

$$2 	imes 1 + 3 	imes (-2) + 4 	imes 2 = 2 - 6 + 8 = 4$$

Question 6

A force $$\vec{F}$$ acts at an angle $$	heta_y$$ from the $$y-$$axis. Which formula gives the $$y-$$component of $$\vec{F}$$?

Accepted Answer

$$F_y = F \cos 	heta_y$$

Question 7

Which of the following best describes the Principle of Transmissibility of forces?

Accepted Answer

A force can be moved anywhere along its line of action without changing the external effect on a rigid body.

Question 8

Given two vectors $$\vec{P}$$ and $$\vec{Q}$$, the scalar (dot) product $$\vec{P} \cdot \vec{Q} is $$zero if:

Accepted Answer

The vectors are perpendicular to each other.

Question 9

A force $$\vec{F}$$ has direction cosines $$\cos 	heta_x = -0.424$$, $$\cos 	heta_y = 0.849$$, $$\cos 	heta_z = 0.318. $$What is the angle $$	heta_x$$ between $$\vec{F}$$ and the $$x-$$axis?

Accepted Answer

$$	heta_x = \cos$$^{-1}$$(-0.424) = 115.09^\circ$$

Question 10

Which of the following is the correct formula for the projection of vector $$\vec{P}$$ onto vector $$\vec{Q}$$?

Accepted Answer

$$\mathrm{proj}_{\vec{Q}} \vec{P} = \dfrac{\vec{P} \cdot \vec{Q}}{|\vec{Q}|}$$

Question 11

A wall section is held by two cables, $$AB$$ and $$AC$$, with tensions $$840\ \mathrm{lb}$$ and $$1200\ \mathrm{lb}$$, respectively. If the resultant force is $$R = \sqrt{(-1440)^2$ + 720^2$ + (-360)^2$}$$, what is the magnitude of $$R$$?

Accepted Answer

$$R = \sqrt{(-1440)^2$ + 720^2$ + (-360)^2$} = 1633\ \mathrm{lb}$$

Question 12

Which of the following is the correct determinant form for the cross product $$\vec{A} 	imes \vec{B}$$, where $$\vec{A} = A_x \vec{i} + A_y \vec{j} + A_z \vec{k}$$ and $$\vec{B} = B_x \vec{i} + B_y \vec{j} + B_z \vec{k}$$?

Accepted Answer

$$\vec{A} 	imes \vec{B} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}$$

Question 13

A block weighing $$155\ \mathrm{kg}$$ rests on an inclined plane at $$32^\circ to $$the horizontal. What is the component of the weight parallel to the plane?

Accepted Answer

$$W_{\parallel} = 155 	imes 9.81 	imes \sin 32^\circ = 805.77\ \mathrm{N}$$

Question 14

Which of the following is NOT a property of the vector (cross) product?

Accepted Answer

It is associative: $$(\vec{A} 	imes \vec{B}) 	imes \vec{C} = \vec{A} 	imes (\vec{B} 	imes \vec{C})$$

Question 15

A force $$\vec{F} is $$defined by its magnitude and two points $$M$$ and $$N on $$its line of action. If $$\vec{d} = (d_x, d_y, d_z) is $$the vector from $$M to$$ $$N$$, what is the unit vector along the line of action?

Accepted Answer

$$\vec{\lambda} = \dfrac{d_x \vec{i} + d_y \vec{j} + d_z \vec{k}}{\sqrt{d_x^2$ + d_y^2$ + d_z^2$}}$$

Question 16

If a force $$\vec{F}$$ has components $$F_x$$, $$F_y$$, and $$F_z$$, which equation gives the magnitude of $$\vec{F}$$?

Accepted Answer

$$F = \sqrt{F_x^2$ + F_y^2$ + F_z^2$}$$

Question 17

A system of forces is in equilibrium if:

Accepted Answer

The resultant of all forces is zero: $$\sum \vec{F} = 0$$

Forces in Three Dimensions: Vector Components and Products

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