Statics: Force Vectors

• This chapter covers force vectors in statics.

Scalars and Vectors

• Scalars have only magnitude, which can be positive or negative.
• Length, area, volume, mass, time, and temperature are examples of scalars.
• Vectors possess magnitude, direction, and sense.
• Position, force, and moment are examples of vectors.
• Vector representation can be symbolic (A) or by magnitude (A or |A|) and direction (angle θ).

Vector Operations

• Multiplying a vector A by a scalar a results in a new vector B where ${\bf{B}} = a{\bf{A}}$.
• The magnitude of B is $B = \left| a \right|A$.
• If a is positive, B has the same direction as A, but if a is negative, B has the opposite direction.

Vector Addition

• The triangle rule and parallelogram law are methods for graphically adding vectors.
• Resultant force ${\bf{F}}_R$ is the sum of individual forces: ${\bf{F}}_R = {\bf{F}}_1 + {\bf{F}}_2$.

Vector Subtraction

• Vector subtraction is defined as ${\bf{A}} - {\bf{B}} = {\bf{A}} + (-{\bf{B}})$.

Cartesian Vectors

• Uses a right-handed coordinate system.
• A vector ${\bf{A}}$ can be expressed in terms of its rectangular components: ${\bf{A}} = A_x{\bf{i}} + A_y{\bf{j}} + A_z{\bf{k}}$.
• $A_x{\bf{i}}$, $A_y{\bf{j}}$, $A_z{\bf{k}}$ are vector components of A in the x, y, z directions while $A_x$, $A_y$, $A_z$ are scalar components of A in the x, y, z directions.

Magnitude of Cartesian Vector

• The magnitude is $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$.

Direction of a Cartesian Vector

• Direction is defined by angles $\alpha$, $\beta$, $\gamma$, and direction cosines $\cos \alpha = \frac{A_x}{A}$, $\cos \beta = \frac{A_y}{A}$, and $\cos \gamma = \frac{A_z}{A}$.
• $\alpha$, $\beta$, and $\gamma$ are the direction angles of A with respect to the x, y, and z axes, respectively.
• The relationship between directional cosines is $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.

Unit Vector

• A unit vector ${\bf{u}}_A = \frac{{\bf{A}}}{A} = \cos \alpha {\bf{i}} + \cos \beta {\bf{j}} + \cos \gamma {\bf{k}}$ defines the direction of A.
• Direction angles can be found by determining the components of the unit vector uA and using the inverse cosine function.
• Cartesian vector representation is ${\bf{A}} = A{\bf{u}}_A = A \cos \alpha {\bf{i}} + A \cos \beta {\bf{j}} + A \cos \gamma {\bf{k}}$ or ${\bf{A}} = A_x{\bf{i}} + A_y{\bf{j}} + A_z{\bf{k}}$.

Addition of Cartesian Vectors

• The resultant vector R from adding vectors A and B is ${\bf{R}} = {\bf{A}} + {\bf{B}} = (A_x + B_x){\bf{i}} + (A_y + B_y){\bf{j}} + (A_z + B_z){\bf{k}}$.

Position Vectors

• A position vector r locates a point in space relative to another point: ${\bf{r}} = x{\bf{i}} + y{\bf{j}} + z{\bf{k}}$.
• The position vector from point A to point B is ${\bf{r}} = (x_B - x_A){\bf{i}} + (y_B - y_A){\bf{j}} + (z_B - z_A){\bf{k}}$.

Force Vector Along a Line

• ${\bf{F}} = F{\bf{u}} = F \frac{{\bf{r}}}{r}$ where u is a unit vector and r is a position vector defining the line's direction, and F is the force magnitude.

Dot Product

• The dot product is defined as ${\bf{A}} \cdot {\bf{B}} = AB \cos \theta$.
• A and B are the magnitudes of vectors A and B, and θ is the angle between them.
• The dot product is commutative: ${\bf{A}} \cdot {\bf{B}} = {\bf{B}} \cdot {\bf{A}}$.
• Multiplication by a scalar: $a({\bf{A}} \cdot {\bf{B}}) = ({\bf{aA}}) \cdot {\bf{B}} = {\bf{A}} \cdot (a{\bf{B}})$.
• The dot product is distributive: ${\bf{A}} \cdot ({\bf{B}} + {\bf{C}}) = ({\bf{A}} \cdot {\bf{B}}) + ({\bf{A}} \cdot {\bf{C}})$.
• In Cartesian form: ${\bf{A}} \cdot {\bf{B}} = A_xB_x + A_yB_y + A_zB_z$.

Applications of the Dot Product

• Allows for finding the angle between two vectors using $\theta = \cos^{-1} \left( \frac{{\bf{A}} \cdot {\bf{B}}}{AB} \right)$.
• It determines the component of a vector in a specified direction: $A_{\parallel} = A \cos \theta = {\bf{A}} \cdot {\bf{u}}$.