1 Vectors for Engineering Mechanics

Vectors and coordinate conventions

A vector represents a quantity with both magnitude and direction. Force, displacement, and moment are vectors; mass and time are scalars. In mechanics, vectors may be written in bold or with an arrow, such as $\vec F$, and the magnitude of $\vec F$ may be written as $|\vec F|$ or simply $F$.

A vector’s components depend on the coordinate axes chosen. State the axes and sign convention before calculating so that the component directions are clear.

Components and unit-vector notation

In a right-handed Cartesian coordinate system, the unit vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ point along the positive $x$-, $y$-, and $z$-axes. Each has magnitude $1$. A vector in three dimensions is represented as

Here, $A_x$, $A_y$, and $A_z$ are scalar components, while $A_x\mathbf{i}$, $A_y\mathbf{j}$, and $A_z\mathbf{k}$ are the corresponding vector components. A negative scalar component means its vector component points opposite the positive direction of that axis. In two dimensions, omit the $z$-component and the $\mathbf{k}$ term.

For a vector of magnitude $A$ directed at angle $\theta$ counterclockwise from the positive $x$-axis in a plane, its components are

For example, a force of $100\,\mathrm{N}$ directed $30^\circ$ above the positive $x$-axis is

The magnitude of a vector can be recovered from its three-dimensional components:

To find the vector from point $P(x_P,y_P,z_P)$ to point $Q(x_Q,y_Q,z_Q)$, subtract the tail coordinates from the head coordinates:

Specifying a direction with a unit vector

A unit vector has magnitude $1$ and specifies direction alone. For a nonzero vector $\vec A$, the unit vector in its direction is

The original vector can therefore be expressed as its magnitude multiplied by its direction unit vector:

When a force’s magnitude and line of action are known, first determine the direction unit vector and then multiply it by the force magnitude.

Addition, subtraction, and scalar multiplication

Add or subtract vectors component by component. For example,

Subtraction is addition of the opposite vector: $\vec A-\vec B=\vec A+(-\vec B)$. The resultant of several forces is their vector sum, found by combining components along the same axis algebraically.

Multiplying a vector $\vec A$ by a scalar $c$ gives a vector $c\vec A$ with magnitude $|c||\vec A|$. A positive value of $c$ preserves the direction, while a negative value reverses it.

Dot product and projections

The dot product of two vectors returns a scalar. It can be calculated from components or from the magnitudes and the angle $\phi$ between the vectors:

The dot product can be used to find a component or projection in a specified direction. For nonzero vectors, if $\vec A\cdot\vec B=0$, the vectors are perpendicular.

Cross product and moments

The cross product returns a vector perpendicular to both input vectors. Its magnitude depends on the angle $\phi$ between them:

Its direction follows the right-hand rule. The order of the vectors matters: $\vec A\times\vec B=-(\vec B\times\vec A)$.

In mechanics, the moment of a force about a point is given by

where $\vec r$ runs from the point about which the moment is taken to the force’s point of application.