Unit Vector Form

Video lesson · from 30:392 worked examples

One lesson video covers all of Vectors, so it opens at 30:39 for this topic — not from the beginning.

Theory

We can express a vector in terms of the standard basis vectors $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ (which have length 1 and point along the x, y, and z axes respectively).

\begin{pmatrix} x \\ y \\ z \end{pmatrix} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}

⚠️ Common Examiner Traps

$i$ , $j$ and $k$ are just the axes: the coefficients are exactly the components of the column vector. Converting between the two forms is purely notation.
Include zero components carefully: a missing $j$ term means that component is zero — do not shuffle the others up.
Convert before adding: if a question mixes column and $i, j, k$ notation, put everything into one form first. Being unable to do this is a common source of lost marks.
Answer in the form asked for: if the question uses $i, j, k$ , give the answer that way.

Worked examples

Example 1

Express in unit vector form:

a) $\begin{pmatrix} 3 \\ 7 \\ 9 \end{pmatrix} = 3\mathbf{i} + 7\mathbf{j} + 9\mathbf{k}$

b) $\begin{pmatrix} 7 \\ -4 \\ -5 \end{pmatrix} = 7\mathbf{i} - 4\mathbf{j} - 5\mathbf{k}$

c) $\begin{pmatrix} -2 \\ 0 \\ 10 \end{pmatrix} = -2\mathbf{i} + 10\mathbf{k}$

Example 2

Express in component form:

d) $-9\mathbf{i} + 6\mathbf{j} + 4\mathbf{k} = \begin{pmatrix} -9 \\ 6 \\ 4 \end{pmatrix}$

e) $12\mathbf{i} + \mathbf{j} - \mathbf{k} = \begin{pmatrix} 12 \\ 1 \\ -1 \end{pmatrix}$

f) $6\mathbf{j} - 2\mathbf{k} = \begin{pmatrix} 0 \\ 6 \\ -2 \end{pmatrix}$