Vectors and geometry

Many quantities in engineering cannot be described by a single number. A force of 500 N acting straight up does something very different from 500 N acting sideways. The wind has a speed and a direction. An electric field at a point has a strength and points somewhere. Quantities like these, with both a size and a direction, are called vectors, and they are the starting point of linear algebra.

This chapter builds the toolkit you will use in statics, dynamics, electromagnetics and fluid mechanics: writing vectors in components, adding and scaling them, measuring angles with the dot product, finding perpendicular directions and moments with the cross product, and describing lines and planes in space. Every idea has a picture, and the pictures are worth learning as carefully as the formulas.

What a vector is

A scalar is a quantity described by one number: mass, temperature, time, energy. A vector has a magnitude and a direction: displacement, velocity, force, acceleration, current density. Geometrically, a vector is an arrow. Its length is the magnitude and the way it points is the direction. Two arrows with the same length and direction are the same vector, wherever they are drawn.

To compute with vectors we set up axes and describe each vector by its components, the distances it travels along each axis. In the plane we write \(\mathbf{u} = (u_1, u_2)\); in space, \(\mathbf{u} = (u_1, u_2, u_3)\). Engineers also write \(\mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j} + u_3\mathbf{k}\), where \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are unit vectors along the \(x\), \(y\) and \(z\) axes. Both notations mean exactly the same thing.

The vector from point \(P\) to point \(Q\) is found by subtracting coordinates: if \(P = (1, 2, 0)\) and \(Q = (4, 6, 0)\), then \(\overrightarrow{PQ} = (3, 4, 0)\). The position vector of a point is the vector from the origin to that point, so its components are just the point's coordinates.

Adding and scaling vectors

Vectors add component by component, and multiplying by a scalar multiplies every component:

Geometrically, to add \(\mathbf{v}\) to \(\mathbf{u}\), place the tail of \(\mathbf{v}\) at the tip of \(\mathbf{u}\). The sum runs from the start of \(\mathbf{u}\) to the end of \(\mathbf{v}\). Doing it in the other order traces the other two sides of a parallelogram and lands in the same place, which is why \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\). The sum is the diagonal of the parallelogram.

Scaling by a positive number stretches or shrinks a vector without changing its direction; scaling by a negative number also reverses it. Subtraction is adding the negative: \(\mathbf{u} - \mathbf{v} = \mathbf{u} + (-1)\mathbf{v}\), which is the arrow from the tip of \(\mathbf{v}\) to the tip of \(\mathbf{u}\) when both start at the same point.

In mechanics, the resultant of several forces acting on a particle is their vector sum. If the resultant is zero, the particle is in equilibrium. That single sentence is the foundation of statics.

Magnitude and unit vectors

The magnitude of a vector comes from Pythagoras, applied once in the plane and twice in space:

A unit vector has magnitude 1. Dividing any non-zero vector by its magnitude gives the unit vector in its direction: \(\hat{\mathbf{u}} = \mathbf{u}/|\mathbf{u}|\). Unit vectors carry direction only, so they are the natural way to turn a known direction and a known size into a vector: \(\mathbf{F} = F\,\hat{\mathbf{u}}\).

The components of a unit vector are the cosines of the angles it makes with the axes, called direction cosines. They satisfy \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\).

This "unit vector times magnitude" step appears in nearly every three-dimensional statics problem. Cable and member directions are always known from geometry; the tensions are the unknowns.

The dot product

The dot product (or scalar product) of two vectors is a number:

Its geometric meaning comes from the law of cosines. If \(\theta\) is the angle between the vectors (placed tail to tail, with \(0 \le \theta \le 180^\circ\)):

Three consequences are used constantly:

• The angle between two vectors: \(\cos\theta = \dfrac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{u}||\mathbf{v}|}\).
• Perpendicular test: two non-zero vectors are perpendicular (orthogonal) exactly when \(\mathbf{u}\cdot\mathbf{v} = 0\).
• Length: \(\mathbf{u}\cdot\mathbf{u} = |\mathbf{u}|^2\).

The sign tells you something too. A positive dot product means the angle is acute (the vectors point roughly the same way); negative means obtuse.

Projections and components along a direction

Often you need the part of a vector that lies along some direction: the component of the weight along a ramp, the part of the wind acting along a runway, the share of a load carried along a member. That part is the projection. The scalar component of \(\mathbf{b}\) along \(\mathbf{a}\) and the vector projection are:

What remains, \(\mathbf{b} - \text{proj}_{\mathbf{a}}\mathbf{b}\), is perpendicular to \(\mathbf{a}\). So every vector splits uniquely into a part along a given direction and a part perpendicular to it. This decomposition is how you separate normal and friction forces on a surface, or tension and shear in a member.

The cross product

The cross product (or vector product) of two vectors in space is another vector. It is defined by a pattern that is easiest to remember as a determinant (Lesson 4 covers determinants fully):

Its geometry is what makes it useful:

• Direction: \(\mathbf{u}\times\mathbf{v}\) is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\), following the right-hand rule. Curl the fingers of your right hand from \(\mathbf{u}\) toward \(\mathbf{v}\); your thumb points along \(\mathbf{u}\times\mathbf{v}\).
• Magnitude: \(|\mathbf{u}\times\mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta\), which is the area of the parallelogram the two vectors span.
• Parallel test: \(\mathbf{u}\times\mathbf{v} = \mathbf{0}\) exactly when the vectors are parallel (or one is zero).

Unlike every product you have met before, the cross product is not commutative. Swapping the order reverses the direction: \(\mathbf{v}\times\mathbf{u} = -\,\mathbf{u}\times\mathbf{v}\). For the unit vectors, \(\mathbf{i}\times\mathbf{j} = \mathbf{k}\), \(\mathbf{j}\times\mathbf{k} = \mathbf{i}\), \(\mathbf{k}\times\mathbf{i} = \mathbf{j}\), and reversing any of these gives a minus sign.

Moments of forces

The turning effect of a force about a point is its moment (also called torque). If \(\mathbf{r}\) is the vector from the point to anywhere on the force's line of action, the moment is:

Here \(d = r\sin\theta\) is the perpendicular distance from the point to the line of action, often called the moment arm. The direction of \(\mathbf{M}\) is the axis the force tends to rotate about, by the right-hand rule.

The scalar triple product

Combining both products gives \(\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\), a number whose absolute value is the volume of the box (parallelepiped) with edges \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\). It is zero exactly when the three vectors lie in one plane. For example, \((1, 0, 0)\cdot\big((0, 2, 0)\times(1, 1, 3)\big) = (1, 0, 0)\cdot(6, 0, -2) = 6\), so those three edges span a box of volume 6. You will see in Lesson 4 that this is a 3×3 determinant.

Lines and planes in space

A line is fixed by a point on it and a direction. If \(\mathbf{p}\) is the position of a point on the line and \(\mathbf{d}\) its direction vector, every point on the line is

This is the parametric form. In components, \(x = p_1 + td_1\), \(y = p_2 + td_2\), \(z = p_3 + td_3\). Think of \(t\) as time and \(\mathbf{d}\) as a velocity: the point slides along the line as \(t\) changes.

A plane is fixed by a point on it and a normal vector \(\mathbf{n}\) perpendicular to it. A point \(\mathbf{r}\) lies on the plane when \(\mathbf{r} - \mathbf{p}\) is perpendicular to \(\mathbf{n}\):

So the coefficients of \(x, y, z\) in a plane's equation are the components of its normal. The plane \(6x + 3y + 2z = 6\) has normal \((6, 3, 2)\).

The distance from a point \(Q\) to a plane is the length of the projection of \(\overrightarrow{PQ}\) onto the normal, for any point \(P\) on the plane: