1.3 Vector algebra and its applications in mechanics

Vector algebra gives you the mathematical tools to represent forces and moments as quantities with both size and direction. Without it, you can't analyze how structures carry loads or determine whether a system is in equilibrium. This topic covers vector operations, component resolution, and how vectors apply directly to force and moment problems in statics.

Vectors and their properties

Vector definition and representation

A vector is a quantity that has both magnitude (size) and direction. You can picture it as an arrow: the length of the arrow tells you "how much," and the way it points tells you "which way."

• The magnitude is the length of the arrow, representing the intensity of the quantity (e.g., a 500 N force, a 3 m/s velocity).
• The direction is the orientation of the arrow, indicating the line of action along which the quantity acts.
• Physical quantities like force, velocity, and displacement are all vectors because they require both magnitude and direction to be fully described.

Compare this to a scalar, which has magnitude only (like mass or temperature). Scalars don't need a direction, so they can't be represented as arrows.

Equality of vectors

Two vectors are equal if and only if they have the same magnitude and the same direction. Their starting points don't matter.

• You can slide a vector anywhere in space (translate it) without changing what it represents, as long as you keep its length and orientation the same.
• This property is called free vector behavior, and it means equal vectors produce identical mechanical effects regardless of where they're drawn.

Vector operations

Vector addition and subtraction

Vector addition combines two or more vectors into a single resultant vector. There are two common geometric methods:

1. Parallelogram law: Place both vectors tail-to-tail. Draw a parallelogram using the two vectors as adjacent sides. The diagonal from the shared tail point to the opposite corner is the resultant.
2. Triangle rule (tip-to-tail): Place the tail of the second vector at the head of the first. The resultant runs from the tail of the first vector to the head of the second.

Both methods give the same result. The triangle rule extends naturally to adding three or more vectors by chaining them tip-to-tail.

Vector subtraction works by adding the negative of a vector. To compute $\vec{A} - \vec{B}$, you reverse the direction of $\vec{B}$ (same magnitude, opposite direction) and then add it to $\vec{A}$ using either method above.

Scalar multiplication and vector products

Scalar multiplication scales a vector's magnitude by a number while keeping (or flipping) its direction.

• Multiply by a positive scalar: the direction stays the same, the magnitude scales.
• Multiply by a negative scalar: the direction reverses, the magnitude scales by the absolute value.

Dot product (scalar product) produces a scalar from two vectors:

$\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta$

where $\theta$ is the angle between the two vectors. The dot product tells you how much one vector projects onto the other. It's commutative ($\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$) and distributive over addition. A dot product of zero means the vectors are perpendicular.

Cross product (vector product) produces a new vector perpendicular to both input vectors:

$\vec{A} \times \vec{B} = |\vec{A}||\vec{B}|\sin\theta\;\hat{n}$

where $\hat{n}$ is the unit vector perpendicular to the plane of $\vec{A}$ and $\vec{B}$, found using the right-hand rule: curl your fingers from $\vec{A}$ toward $\vec{B}$, and your thumb points in the direction of the result.

• The cross product is anticommutative: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$.
• It's distributive over addition.
• A cross product of zero means the vectors are parallel.

The cross product is especially important in statics because it's how you calculate moments.

Resolving vectors into components

Vector resolution

Resolving a vector means breaking it into perpendicular parts along coordinate axes. This converts a single angled vector into two (or three) simpler pieces you can work with algebraically.

For a 2D vector with magnitude $F$ at angle $\theta$ from the positive x-axis:

1. x-component: $F_x = F\cos\theta$
2. y-component: $F_y = F\sin\theta$

In 3D, you also get a z-component. Each component is the projection of the vector onto that axis. Once resolved, you can write the vector in component form as $\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}$, where $\hat{i}$, $\hat{j}$, and $\hat{k}$ are unit vectors along the x, y, and z axes.

Calculating magnitude and direction from components

Going the other direction, you can reconstruct magnitude and direction from known components.

Magnitude uses the Pythagorean theorem:

• 2D: $|\vec{F}| = \sqrt{F_x^2 + F_y^2}$
• 3D: $|\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$

Direction in 2D:

$\theta = \text{atan2}(F_y, F_x)$

Use $\text{atan2}$ rather than plain $\arctan$ because it correctly handles all four quadrants. In 3D, direction is typically described by direction cosines: the cosine of the angle between the vector and each coordinate axis.

$\cos\alpha = \frac{F_x}{|\vec{F}|}, \quad \cos\beta = \frac{F_y}{|\vec{F}|}, \quad \cos\gamma = \frac{F_z}{|\vec{F}|}$

A useful check: $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.

Vector algebra in physics problems

Forces as vectors

Forces are naturally represented as vectors: the magnitude is the force intensity (in Newtons), and the direction is the line of action.

To find the resultant force on an object, add all force vectors. In practice, you'll almost always resolve each force into components first, then sum the components separately:

1. Resolve every force into x, y (and z) components.
2. Sum all x-components: $R_x = \sum F_x$
3. Sum all y-components: $R_y = \sum F_y$
4. Find the resultant magnitude: $R = \sqrt{R_x^2 + R_y^2}$
5. Find the resultant direction: $\theta_R = \text{atan2}(R_y, R_x)$

Equilibrium occurs when the resultant force is zero. This gives you the equations:

• 2D: $\sum F_x = 0$ and $\sum F_y = 0$
• 3D: $\sum F_x = 0$, $\sum F_y = 0$, and $\sum F_z = 0$

These equilibrium equations are the foundation of every statics problem you'll solve.

Moments as vectors

A moment (or torque) measures a force's tendency to cause rotation about a point. The moment vector is calculated using the cross product:

$\vec{M} = \vec{r} \times \vec{F}$

where $\vec{r}$ is the position vector from the point of rotation to the point where the force is applied, and $\vec{F}$ is the force vector.

The magnitude of the moment is:

$|\vec{M}| = |\vec{r}||\vec{F}|\sin\theta$

where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$. The term $|\vec{r}|\sin\theta$ is the perpendicular distance (moment arm) from the line of action of the force to the rotation point. A larger moment arm means a greater rotational effect for the same force.

The direction of $\vec{M}$ (found via the right-hand rule) tells you the axis of rotation and the sense (clockwise vs. counterclockwise).

Moment equilibrium requires:

• 2D: $\sum M = 0$ (about any point)
• 3D: $\sum M_x = 0$, $\sum M_y = 0$, and $\sum M_z = 0$

Combined with force equilibrium, these conditions give you the complete set of equations needed to solve static equilibrium problems.