Back
Vectors
A-Level H2 Math
Basics
Unit VectorMidpoint TheoremRatio Theorem
$\hat{v} = \dfrac{\mathbf{v}}{|\mathbf{v}|}$$\overrightarrow{OM} = \dfrac{\mathbf{a}+\mathbf{b}}{2}$$\overrightarrow{OM} = \dfrac{\lambda\mathbf{a} + \mu\mathbf{b}}{\lambda + \mu}$
Dot Product & Cross Product
Dot Product
  • $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$
  • $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$
  • $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
  • $\mathbf{a} \cdot (\mathbf{b}+\mathbf{c}) = \mathbf{a}\cdot\mathbf{b} + \mathbf{a}\cdot\mathbf{c}$
  • $\mathbf{a} \cdot \mathbf{b} = 0 \implies \mathbf{a} \perp \mathbf{b}$
  • $\mathbf{a} \cdot \mathbf{b} > 0 \implies$ same direction
  • $\mathbf{a} \cdot \mathbf{b} < 0 \implies$ opposite direction
Cross Product
  • $\mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta\,\hat{n}$
  • $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$
  • $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$
  • $\mathbf{a} \times (\mathbf{b}+\mathbf{c}) = \mathbf{a}\times\mathbf{b} + \mathbf{a}\times\mathbf{c}$
  • $(\mathbf{b}+\mathbf{c}) \times \mathbf{a} = \mathbf{b}\times\mathbf{a} + \mathbf{c}\times\mathbf{a}$
  • $\mathbf{a} \times \mathbf{b} = \mathbf{0} \implies \mathbf{a} \parallel \mathbf{b}$
  • In particular, $\mathbf{a} \times \mathbf{a} = \mathbf{0}$
Lines & Planes
Line
$\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$
$\dfrac{x-a_1}{d_1} = \dfrac{y-a_2}{d_2} = \dfrac{z-a_3}{d_3}$
Plane
$\mathbf{r} \cdot \mathbf{n} = d$
$ax + by + cz = d$
$\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}_1 + \mu\mathbf{d}_2$
Projections & Distances
DescriptionFormula
Projection of $\overrightarrow{PQ}$ onto line (direction $\mathbf{d}$)$\dfrac{|\overrightarrow{PQ} \cdot \mathbf{d}|}{|\mathbf{d}|}$
Projection of $\overrightarrow{PQ}$ onto plane (normal $\mathbf{n}$)$\dfrac{|\overrightarrow{PQ} \times \mathbf{n}|}{|\mathbf{n}|}$
Distance from point to plane (normal $\mathbf{n}$)$\dfrac{|\overrightarrow{PQ} \cdot \mathbf{n}|}{|\mathbf{n}|}$
Distance between two parallel planes$\dfrac{|d_1 - d_2|}{|\mathbf{n}|}$
Angles
BetweenFormula
Two lines$\cos\theta = \dfrac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}$
Line and plane$\sin\theta = \dfrac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$
Two planes$\cos\theta = \dfrac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{|\mathbf{n}_1||\mathbf{n}_2|}$

Formulas for the Singapore A-Level H2 Mathematics syllabus