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# Statics ## Chapter 2 ### Force Vectors #### 2.1 Scalars and Vectors **Scalar:** A quantity characterized by a positive or negative number. **Vector:** A quantity that has magnitude, direction, and sense. **Parallelogram Law:** The sum of **A** and **B** is **R**. $$ \mathbf{R} = \mathbf{A}...

# Statics ## Chapter 2 ### Force Vectors #### 2.1 Scalars and Vectors **Scalar:** A quantity characterized by a positive or negative number. **Vector:** A quantity that has magnitude, direction, and sense. **Parallelogram Law:** The sum of **A** and **B** is **R**. $$ \mathbf{R} = \mathbf{A} + \mathbf{B} $$ **Triangle Rule:** Vector addition is commutative. $$ \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} $$ **Vector Subtraction:** $$ \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B}) $$ #### 2.2 Vector Operations **Scalar Multiplication and Division:** $$ \mathbf{B} = a\mathbf{A} $$ * Magnitude: $B = |a|A$ * If $a$ is positive, **B** has the same direction as **A**. * If $a$ is negative, **B** is opposite to **A**. **Vector Addition:** All forces acting at point $A$ can be added together resulting in the resultant force. $$ \mathbf{F}_R = \sum \mathbf{F} = \mathbf{F}_1 + \mathbf{F}_2 $$ ### 2.3 Cartesian Vectors **Right-Handed Coordinate System:** The $z$-axis is oriented so that the thumb of the right hand points along the $z$-axis when the fingers are curled from the $x$-axis to the $y$-axis. **Rectangular Components of a Vector:** $$ \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k} $$ Magnitude: $$ A = \sqrt{A_x^2 + A_y^2 + A_z^2} $$ **Direction:** $$ \begin{aligned} & \cos \alpha = \frac{A_x}{A} \\ & \cos \beta = \frac{A_y}{A} \\ & \cos \gamma = \frac{A_z}{A} \end{aligned} $$ where $\alpha$, $\beta$, and $\gamma$ are the direction angles of **A** (measured between the tail of **A** and the positive $x, y, z$ axes). **Directional Cosines:** $$ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 $$ **Unit Vector:** $$ \mathbf{u}_A = \frac{\mathbf{A}}{A} = \frac{A_x}{A}\mathbf{i} + \frac{A_y}{A}\mathbf{j} + \frac{A_z}{A}\mathbf{k} $$ Also, $$ \mathbf{u}_A = \cos \alpha \mathbf{i} + \cos \beta \mathbf{j} + \cos \gamma \mathbf{k} $$ So, the vector **A** can be written as $$ \mathbf{A} = A\mathbf{u}_A = A\cos \alpha \mathbf{i} + A\cos \beta \mathbf{j} + A\cos \gamma \mathbf{k} $$ **Cartesian Vector Addition:** $$ \begin{aligned} \mathbf{R} &= \mathbf{A} + \mathbf{B} = (A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}) + (B_x \mathbf{i} + B_y \mathbf{j} + B_z \mathbf{k}) \\ &= (A_x + B_x)\mathbf{i} + (A_y + B_y)\mathbf{j} + (A_z + B_z)\mathbf{k} \\ \mathbf{R} &= R_x\mathbf{i} + R_y\mathbf{j} + R_z\mathbf{k} \end{aligned} $$ Where $$ \begin{aligned} R_x &= A_x + B_x \\ R_y &= A_y + B_y \\ R_z &= A_z + B_z \end{aligned} $$ ### 2.4 Position Vectors **Position Vector:** A fixed vector which locates a point in space relative to another point. $$ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} $$ **Position Vector Directed From Point A to Point B:** $$ \mathbf{r} = (x_B - x_A)\mathbf{i} + (y_B - y_A)\mathbf{j} + (z_B - z_A)\mathbf{k} $$ We can also write $$ \mathbf{r} = r\mathbf{u} $$ Where $$ \mathbf{u} = \frac{\mathbf{r}}{r} = \frac{(x_B - x_A)}{r}\mathbf{i} + \frac{(y_B - y_A)}{r}\mathbf{j} + \frac{(z_B - z_A)}{r}\mathbf{k} $$ Here, $\mathbf{u}$ is a unit vector in the direction of $\mathbf{r}$. ### 2.5 Force Vector Directed Along a Line **Force Vector:** Characterized by its magnitude and a direction $\mathbf{u}$ $$ \mathbf{F} = F\mathbf{u} $$ Also, the unit vector $\mathbf{u}$ can be determined from the position vector $\mathbf{r}$. $$ \mathbf{u} = \frac{\mathbf{r}}{r} = \frac{(x_B - x_A)}{r}\mathbf{i} + \frac{(y_B - y_A)}{r}\mathbf{j} + \frac{(z_B - z_A)}{r}\mathbf{k} $$ So, $$ \mathbf{F} = F\mathbf{u} = F\frac{\mathbf{r}}{r} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k} $$ ### 2.6 Dot Product The dot product of vectors **A** and **B** is defined as $$ \mathbf{A} \cdot \mathbf{B} = A B \cos \theta $$ Where * $A$ and $B$ represent the magnitudes of **A** and **B** * $\theta$ is the angle between the tails of **A** and **B** **Laws of Operation:** * Commutative law: $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$ * Multiplication by a scalar: $a(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot a\mathbf{B})$ * Distributive law: $\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = (\mathbf{A} \cdot \mathbf{B}) + (\mathbf{A} \cdot \mathbf{C})$ **Cartesian Vector Formulation:** $$ \begin{aligned} \mathbf{A} \cdot \mathbf{B} &= (A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}) \cdot (B_x \mathbf{i} + B_y \mathbf{j} + B_z \mathbf{k}) \\ &= A_x B_x + A_y B_y + A_z B_z \end{aligned} $$ Using above equations, the angle $\theta$ between two vectors can be determined from $$ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{AB} = \frac{A_x B_x + A_y B_y + A_z B_z}{\sqrt{A_x^2 + A_y^2 + A_z^2}\sqrt{B_x^2 + B_y^2 + B_z^2}} $$ **Determining the Direction Angles of a Vector:** $$ \begin{aligned} & A_x = A \cos \alpha = \mathbf{A} \cdot \mathbf{i} \\ & A_y = A \cos \beta = \mathbf{A} \cdot \mathbf{j} \\ & A_z = A \cos \gamma = \mathbf{A} \cdot \mathbf{k} \end{aligned} $$ **Finding the projection of a vector onto a line (or another vector)** The projection of vector **A** onto line $a$ is defined as $$ A_a = A \cos \theta = \mathbf{A} \cdot \mathbf{u} $$ Where $\mathbf{u}$ is a unit vector that defines the direction of line $a$. In general, $A_a$ represents the magnitude of the projection of **A** onto line $a$. The projection vector $\mathbf{A}_a$ can be written as $$ \mathbf{A}_a = A_a \mathbf{u} = (\mathbf{A} \cdot \mathbf{u})\mathbf{u} $$