Adding vectors and multiplying by scalars

By the end of this lesson

By the end you can add and subtract vectors, multiply them by numbers, and say what each of these does to the arrows in a diagram.

Adding moves

Think of a vector as a move. Walk 2 east and 1 north, then 3 east and 4 north. Altogether you've gone $2 + 3 = 5$ east and $1 + 4 = 5$ north. That's all vector addition is: add the matching components.

$\begin{pmatrix} 2 \\ 1 \end{pmatrix} + \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 + 3 \\ 1 + 4 \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}$

The same works in 3D (three rows) and in $\mathbf{i}, \mathbf{j}, \mathbf{k}$ form, where you collect like terms: $(2\mathbf{i} + \mathbf{j}) + (3\mathbf{i} + 4\mathbf{j}) = 5\mathbf{i} + 5\mathbf{j}$ .

Subtracting works the same way, row by row: $\begin{pmatrix} 5 \\ 5 \end{pmatrix} - \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ .

Multiplying by a number $k$ (a scalar) multiplies every component: $3\begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 6 \\ 3 \end{pmatrix}$ , "the same move, three times over".

Worked example: Example 1: working with components

$\mathbf{a} = \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}$ . Find (a) $\mathbf{a} + \mathbf{b}$ , (b) $2\mathbf{a} - \mathbf{b}$ , (c) $-\tfrac12\mathbf{b}$ .

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