Solving Problems with Vectors

Age 16 to 18
Article by Alison Kiddle

Published 2014

Many students are often reluctant to tackle questions using vectors. I think this is partly because often vectors is not taught until quite a way through a school maths course, so they are unfamiliar. This short article aims to highlight some of the powerful techniques that can be used to solve problems involving vectors, and to encourage you to have a go at such problems to become more familiar with vector properties and applications.

So what are vectors? When we first meet them, it's often in the context of transformations - a translation can be expressed as a vector telling us how far something is translated to the right (or left) and up (or down). Confusion can strike when we come across vectors being used to indicate absolute position relative to an origin as well as showing a direction. Then we may be informed that a vector is "simply" a quantity that has both magnitude and direction (unlike a scalar which only has magnitude).

It is helpful to separate out some of these ideas about vectors in order to make sense of things. For me, diagrams make it much easier to make sense of what is going on - I can represent a position vector as a point on the diagram with a line segment coming from the origin. Direction vectors just become line segments joined onto other vectors, with a helpful arrow to remind me that $\mathbf{a}$ and $\mathbf{-a}$ are in opposite directions! Sometimes it's useful to draw on lines parallel and perpendicular to my coordinate axes so I can make sense of the x and y components of a vector.

Given a vector problem, a quick sketch can help you to see what's going on, and the act of transferring the problem from the written word to a diagram can often give you some insight that will help you to find a solution. Start by solving vector problems in two dimensions - it's easier to draw the diagrams - and then move on to three dimensions. (For four or more dimensions, it becomes more difficult to visualise!)

What to do when you get stuck
Here is a brief checklist of ideas to think about if you are stuck on a vector question, and drawing a diagram hasn't helped.

Parallel vectors
Do you know about any parallel lines? Vector questions can often be about geometrical shapes like trapezia, rhombuses or parallelograms. If two vectors are parallel, it can be really useful to express one in terms of the other - if $\mathbf{a}$ and $\mathbf{b}$ are parallel, try writing $\mathbf{b}=k\mathbf{a}$ for some constant $k$.

Scalar products
Scalar products are immensely useful! Sometimes if you're at a loss to know what to do with vectors and vector equations, it's worth just taking the scalar product of the whole equation with one of your vectors and seeing what you end up with. Remember, $\mathbf{a}.\mathbf{a}=|a|^2$, and if two vectors are perpendicular, their scalar product is $0$.

Magnitude and direction
Some vector problems involve a vector function which tells you how an object's position changes in time, for example. Working out how the magnitude and direction change over time can help you to picture the situation.

Vector equation of a line
Some students are intimidated by the vector equation of a line when they first meet it. We are very used to expressing lines using cartesian geometry in the form $y=mx+c$ and other variants. The vector equation of a line is no more complicated really, it's just a case of getting used to it. In simple terms, lines are represented using vectors by specifying a point on the line with a position vector, and then using a direction vector to specify the direction of the line. In the same way that $y=mx+c$ specifies a line that passes through $(0,c)$ and has gradient $m$, the vector equation $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$ specifies a line that passes through the point with position vector $\mathbf{a}$ in the direction of $\mathbf{b}$.

A final word on notation; in type, vectors are indicated by bold type. In handwriting, it is a convention to underline vectors and leave scalars (such as the constants $k$ and $\lambda$ above without underlining. The Greek letters $\lambda$ and $\mu$ are often used as constants in vector equations, so why not get into the habit of using them for yourself?