2025 JC 1 H2 Math Vectors 1 Summary Flashcards

A comprehensive set of vocabulary flashcards covering the fundamental formulas and properties of vectors, including scalar and vector products, projections, and geometric distances for H2 Mathematics.

Scalar Product (Dot Product)

$$a \cdot b = |a||b|\cos(\theta)$$

Scalar Product Rule for Self-Product

$$a \cdot a = |a|^2$$

Condition for Perpendicularity (Scalar Product)

If $a$ and $b$ are perpendicular, then $a \cdot b = 0$. Converesly, if $a \cdot b = 0$, then either $a = 0$, $b = 0$, or $a$ and $b$ are perpendicular.

Scalar Product of Standard Basis Vectors

$i \cdot i = j \cdot j = k \cdot k = 1$ and $i \cdot j = 0$, $j \cdot k = 0$, $k \cdot i = 0$

Magnitude of Vector OP

$|OP| = \sqrt{x^2 + y^2 + z^2}$ for $OP = xi + yj + zk$

Unit Vector

A vector in the direction of $u$ defined as $\hat{u} = \frac{u}{|u|}$, such that $u = |u|\hat{u}$

Length of Projection of a on b

$$|a \cdot \hat{b}| = \frac{|a \cdot b|}{|b|}$$

Ratio Theorem

Formula to find the position vector of point $P$ dividing segment $AB$ in ratio $\lambda : \mu$: $OP = \frac{\mu OA + \lambda OB}{\lambda + \mu}$

Vector Product (Cross Product)

$a \times b = |a||b|\sin(\theta)\hat{n}$, where $\hat{n}$ is determined by the right hand rule.

Vector Product Anti-commutativity

$$a \times b = -(b \times a)$$

Vector Product Rule for Self-Product

$$a \times a = 0$$

Shortest Distance from point C to Line AB

Perpendicular distance = $|AC \times \hat{d}|$ where $\hat{d}$ is the unit direction vector of the line ($\frac{d}{|d|}$).

Vector Product of Standard Basis Vectors

$i \times i = 0$, $j \times j = 0$, $k \times k = 0$ and $i \times j = k$, $j \times k = i$, $k \times i = j$

Angle between 2 vectors (Cosine Formula)

$$\theta = \cos^{-1}\left(\frac{AB \cdot AC}{|AB||AC|}\right)$$

Area of Triangle ABC

$$\frac{1}{2} |AB \times AC| = \frac{1}{2} |BA \times BC| = \frac{1}{2} |CA \times CB|$$

Area of Parallelogram ABPC

$$|AB \times AC| = 2 \times \text{Area of triangle } ABC$$