​​Representing vectors

In a nutshell

You can represent vectors in column vector form $$\begin{pmatrix}a\\b\end{pmatrix}$$. This shows its displacement relative to the $$x$$- and $$y$$- axes. You can add or subtract column vectors and multiply vectors by a scalar. A unit vector is a vector with a magnitude of 1. Unit vectors in the positive $$x$$​ and $$y$$​ directions are called '$$\textbf{i}$$' and '$$\textbf{j}$$', repesectively.

Definitions

Column vectors

A vector describes a displacement between points. In a two-coordinate system, you can write this displacement relative to the $$x$$- and $$y$$-axes in column vector form. $$\textbf{v}=\begin{pmatrix}a\\b\end{pmatrix}$$ means move $$a$$ units along the positive $$x$$​-axis, then $$b$$​ units along the positive $$y$$​-axis.​

Example 1

Write $$\textbf{a}$$​ in column vector form.

The vectors takes $$2$$ steps right then $$3$$ steps up, so:

$$\textbf{a}=\underline{\begin{pmatrix}2\\3\end{pmatrix}}$$​​

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Adding and subtracting vectors

To add and subtract column vectors, work out the operations row by row:

​$$\boxed{\begin{pmatrix}a\\b\end{pmatrix}+\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a+c\\b+d\end{pmatrix}}$$​​

Multiplying a vector by a scalar

To multiply a vector by a scalar $$k$$, multiply each term by the scalar:

​$$\boxed{k\begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}ka\\kb\end{pmatrix}}$$​​

Example 2

If $$\textbf{a}=\begin{pmatrix}2\\1\end{pmatrix}$$ and $$\textbf{b}=\begin{pmatrix}3\\1\end{pmatrix}$$, find $$\textbf{a}+\textbf{b}$$.

​$$\begin{aligned}\textbf{a}+\textbf{b}=\begin{pmatrix}2\\1\end{pmatrix}+\begin{pmatrix}3\\1\end{pmatrix}=\begin{pmatrix}2+3\\1+1\end{pmatrix}=\underline{\begin{pmatrix}5\\2\end{pmatrix}}\end{aligned}$$​​​

Example 3

If $$\textbf{a}=\begin{pmatrix}2\\1\end{pmatrix}$$, find $$-3\textbf{a}$$.

$$-3\textbf{a}=-3\begin{pmatrix}2\\1\end{pmatrix}=\begin{pmatrix}-3\times2\\-3\times1\end{pmatrix}=\underline{\begin{pmatrix}-6\\-3\end{pmatrix}}$$​​

Unit vectors

A unit vector is a vector with a magnitude of $$1$$​. For the positive $$x$$ and $$y$$ directions, the unit vectors are commonly referred to as $$\textbf{i}$$ and $$\textbf{j}$$, respectively.​

​​$$\boxed{\begin{aligned}\textbf{i}&=\begin{pmatrix}1\\0\end{pmatrix}\\\\\textbf{j}&=\begin{pmatrix}0\\1\end{pmatrix}\end{aligned}}$$​​

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Any two-dimensional vector $$\textbf{v}$$ can be written in terms of $$\textbf{i}$$​ and $$\textbf{j}$$, or component form:

$$\boxed{\textbf{v}=\begin{pmatrix}a\\b\end{pmatrix}=a\textbf{i}+b\textbf{j}}$$​​

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Example 4

Draw a diagram to represent the vector $$4\textbf{i}-3\textbf{j}$$ and rewrite it in column vector form.

The diagram would be:

In column vector form:

$$4\textbf{i}-3\textbf{j}=\underline{\begin{pmatrix}4\\-3\end{pmatrix}}$$​​

Example 5

If $$\textbf{a}=2\textbf{i}+5\textbf{j}$$ and $$\textbf{b}=\textbf{i}-3\textbf{j}$$, find $$\textbf{a}+2\textbf{b}$$, giving your answer in terms of $$\textbf{i}$$ and $$\textbf{j}$$.​

$$\begin{aligned}\textbf{a}+2\textbf{b}&=(2\textbf{i}+5\textbf{j})+2(\textbf{i}-3\textbf{j})\\&=(2\textbf{i}+5\textbf{j})+(2\textbf{i}-6\textbf{j})\\&=\underline{4\textbf{i}-\textbf{j}}\end{aligned}$$​​

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