Further vectors – further maths revision notes to download as PDF

Starter activity

A drone is flying over a field. A radio mast stands at the origin of a coordinate system with the \(z\)-axis pointing straight up. At one moment the drone is at the point \((30,40,12)\) metres. How far is it from the mast? How far is it from the vertical line through the mast? If the drone flies in the direction \((3,4,0)\), is it moving towards the mast or away from it? Discuss how you could answer each question using a number calculated from two vectors, and decide whether you would need a product that gives a number or one that gives a vector.

Note: three ways to describe a line

A line in three dimensions can be described by a vector equation \(\mathbf r=\mathbf a+\lambda\mathbf d\), by parametric equations \(x=a_1+\lambda d_1\), \(y=a_2+\lambda d_2\), \(z=a_3+\lambda d_3\), or by the symmetric Cartesian form \(\dfrac{x-a_1}{d_1}=\dfrac{y-a_2}{d_2}=\dfrac{z-a_3}{d_3}\). Unlike in two dimensions, a line cannot be described by a single linear equation, because one equation in \(x\), \(y\) and \(z\) describes a plane. The symmetric form is really two equations, which describe two planes whose intersection is the line. If a component of the direction vector is zero, the symmetric form needs adjusting: for \(\mathbf d=(2,0,3)\) you would write \(\dfrac{x-a_1}2=\dfrac{z-a_3}3\) with \(y=a_2\).

Applications of vectors in three dimensions

Vectors in three dimensions are the language of engineering and computer graphics. Architects use planes to describe roofs and floors, and the angle between two planes tells them how steeply a roof meets a wall. Air traffic controllers monitor the shortest distance between the lines representing flight paths, and they alert pilots when two aircraft will pass too close. Game designers use the dot product to decide whether a surface faces towards a light source, which determines how bright it should appear, and they use the cross product to compute the normal of every triangle in a model. In physics the moment of a force about a point, the force on a charged particle in a magnetic field, and the angular momentum of a spinning body are all given by cross products.

Thinking geometrically before calculating

Many mistakes in three-dimensional vector geometry come from rushing straight into algebra without a picture. Before you start, decide what kind of objects you are working with: a point, a line or a plane. A point has three coordinates. A line is described by a point and one direction. A plane is described by a point and two directions, or by a point and one normal direction. Next, ask what relationship the question is about. If it asks for an angle, you want the dot product. If it asks for a perpendicular direction or an area, you want the cross product. If it asks for a distance, you want one of the three distance formulae. Finally, ask whether the answer should be a point, a number or an equation, and give it in that form. This simple checklist turns an apparently complicated problem into a short sequence of familiar steps.

Did you know?

The vector products used today were invented in the nineteenth century. William Rowan Hamilton, an Irish mathematician, discovered quaternions in 1843 while walking by the Royal Canal in Dublin, and he carved the formula for them into Broom Bridge. The cross product and dot product were extracted from the quaternions by Josiah Willard Gibbs and Oliver Heaviside around 1880, and they became the standard tools of physicists. Today the quaternions themselves are used inside every smartphone and video game, because they describe three-dimensional rotations without the problem of gimbal lock.