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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