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A vector has magnitude and direction. In AS Mathematics you write vectors as column vectors or with the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\), and you add, subtract and scale them component by component. Vectors appear in Pure (Paper 1) and in Mechanics (Paper 2), usually as the first part of a longer question.
Key facts
\(a\mathbf{i} + b\mathbf{j} + c\mathbf{k} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}\)
\(\mathbf{i}\),\(\mathbf{j}\),\(\mathbf{k}\): unit vectors along the \(x\)-, \(y\)- and \(z\)-axes
Add or subtract component by component
\(\lambda\begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} \lambda a \\ \lambda b \\ \lambda c \end{pmatrix}\)
Parallel: \(\mathbf{b} = \lambda\mathbf{a}\) for some scalar \(\lambda\)
Equal vectors: all components equal
Notes
What a vector is
A scalar has size only (e.g. mass, speed). A vector has size and direction (e.g. displacement, velocity, force).
Notation: \(\mathbf{a}\) in print, \(\underline{a}\) (underlined) in handwriting, or \(\overrightarrow{AB}\) for the vector from \(A\) to \(B\)
\(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are unit vectors (magnitude 1) in the directions of the \(x\)-, \(y\)- and \(z\)-axes.
Component form
diagram
\(a\mathbf{i} + b\mathbf{j} + c\mathbf{k} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}\). In two dimensions there is no \(\mathbf{k}\) term.
\(3\mathbf{i} + 2\mathbf{j} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\): 3 units in the \(\mathbf{i}\) direction, then 2 in the \(\mathbf{j}\) direction.
Arithmetic
Add or subtract matching components. To multiply by a scalar, multiply every component.
\(-\mathbf{a}\) has the same magnitude as \(\mathbf{a}\) but the opposite direction. \(\mathbf{0}\) is the zero vector.
\(\lambda\mathbf{a}\) is parallel to \(\mathbf{a}\); if \(\lambda \lt 0\) it points the opposite way.
Equating components
\(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are independent: no combination of two of them makes the third. So if \(p\mathbf{i} + q\mathbf{j} + r\mathbf{k} = 5\mathbf{i} - 2\mathbf{j} + \mathbf{k}\), then \(p = 5\), \(q = -2\) and \(r = 1\)
This turns one vector equation into simultaneous equations; a spare component can be used as a check.
How to answer each type of question
M = method mark, A = answer mark, B = a mark on its own.
Combine vectors in component form
2 marksGrade E
Multiply each vector by its scalar, component by component.
Add or subtract matching components, using brackets for negatives.
Collect the \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) components on the left-hand side.
Equate each component with the right-hand side to get simultaneous equations.
Solve two of them and check the solution in the third.
Example. Find the values of the constants \(p\) and \(q\) such that \(p(2\mathbf{i} + \mathbf{j} - \mathbf{k}) + q(\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}) = 7\mathbf{i} + 7\mathbf{j} - 6\mathbf{k}\)