Parallel Vectors If $\mathbf{a}\parallel \mathbf{b}$, then $\mathbf{a}=k\mathbf{b}$, where • $\mathbf{a}\cdot \mathbf{b}=\left| \mathbf{a} \right|\left| \mathbf{b} \right|$ • $\mathbf{a}\times \mathbf{b}=0$ Perpendicular Vectors If $\mathbf{a}\bot \mathbf{b}$, • $\mathbf{a}\cdot \mathbf{b}=0$ • $\left| \mathbf{a}\times \mathbf{b} \right|=\left| \mathbf{a} \right|\left| \mathbf{b} \right|$ Collinear Points If points $A$, $B$ and $C$ are collinear then$\overrightarrow{AB}=k\overrightarrow{BC}$, where \[k\subseteq \mathbb{R}\] and $B$ is a common point.

Vector Tests and Essential Skills

Parallel vectors and collinear points

\[\mathbf a\parallel\mathbf b\iff \mathbf a=\lambda\mathbf b\ \text{for some scalar}\hspace{0.5em}\lambda\quad(\mathbf b\ne\mathbf0).\qquad A,B,C\text{ are collinear}\iff\overrightarrow{AB}=\lambda\overrightarrow{AC}.\]

Relationship	Reliable test for nonzero vectors
Parallel	$\mathbf a=\lambda\mathbf b$ or $\mathbf a\times\mathbf b=\mathbf0$
Perpendicular	$\mathbf a\cdot\mathbf b=0$

Use this self-audit to check that each relationship can be recognized, tested and computed. The empty squares are checklist markers, not answer blanks.

Relationship	I can…
Point and line	□ test whether a point lies on a line. □ find the foot and shortest distance.
Point and plane	□ test whether a point lies on a plane. □ find the foot and shortest distance.
Two lines	□ distinguish coincident, parallel, intersecting and skew lines. □ find their acute angle or shortest distance.
Line and plane	□ distinguish contained, parallel, intersecting and perpendicular cases. □ find the acute angle and point of intersection.
Two planes	□ distinguish coincident, parallel and intersecting planes. □ find their acute angle, distance or line of intersection.
Vector basics	□ calculate magnitudes and unit vectors. □ use components and vector algebra laws.
Ratios	□ use collinearity and internal-division formulae. □ interpret the ratio orientation correctly.
Scalar product	□ test perpendicularity and find an angle. □ distinguish signed projection from projection length.
Vector product	□ test parallelism and determine orientation. □ find parallelogram and triangle areas.
Equations	□ form line equations from one or two points. □ form plane equations from directions or a normal.
Problem solving	□ draw a labelled sketch. □ use a graphing calculator where numerical solving is appropriate.

State nonzero-vector conditions before dividing by a magnitude.
Keep absolute values in distance and acute-angle formulae.
Use one consistent parameter per line and check all components.
Mark only genuine intersections; skew lines do not intersect.
Verify the final point or direction in the original equations.

Create a free account to practise similar questions.

Create free account Sign in Get the H2 Math course · from S$39.98/mo H2 Math tuition

Need help? Join our JC Math tuition classes.

Learn more

Relationship	Reliable test for nonzero vectors
Parallel	\(\mathbf a=\lambda\mathbf b\) or \(\mathbf a\times\mathbf b=\mathbf0\)
Perpendicular	\(\mathbf a\cdot\mathbf b=0\)