H2 Maths Notes (JC 1-2): 3.1) Basic Properties of Vectors

Before you revise\ Keep vector diagrams clean—label initial points, direction arrows, and magnitudes. Many marks are lost to orientation errors or forgetting column-vector format.

Core Definitions

• A vector \( \vec{v} \) has magnitude \( \lVert \vec{v} \rVert \) and direction; in component form \( \vec{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix} \).
• Position vector of point \( A(x, y, z) \) is \( \vec{OA} = \begin{pmatrix} x \ y \ z \end{pmatrix} \).
• Magnitude: \( \lVert \vec{v} \rVert = \sqrt{v_1^2 + v_2^2 + v_3^2} \).
• Unit vector: \( \hat{v} = \frac{\vec{v}}{\lVert \vec{v} \rVert} \).

Vector Operations

• Addition/subtraction component-wise: \( \vec{a} \pm \vec{b} = \begin{pmatrix} a_1 \pm b_1 \ a_2 \pm b_2 \ a_3 \pm b_3 \end{pmatrix} \).
• Scalar multiplication: \( k \vec{a} = \begin{pmatrix} ka_1 \ ka_2 \ ka_3 \end{pmatrix} \).
• Parallel vectors satisfy \( \vec{a} = k \vec{b} \).
• Midpoint of segment \( AB \): \( \vec{OM} = \tfrac{1}{2}(\vec{OA} + \vec{OB}) \).

Example -- Midpoint

If \( A(2, -1, 5) \) and \( B(-4, 7, 1) \), then

\[ \vec{OM} = \frac{1}{2}\begin{pmatrix} 2 + (-4) \ -1 + 7 \ 5 + 1 \end{pmatrix} = \begin{pmatrix} -1 \ 3 \ 3 \end{pmatrix}. \]

Lines in Vector Form

• A line through point \( A \) with direction vector \( \vec{d} \) is \( \vec{r} = \vec{OA} + \lambda \vec{d} \).
• Convert to parametric equations: \( x = x_A + \lambda d_1 \), etc.
• Two points determine a line: direction vector \( \vec{d} = \vec{OB} - \vec{OA} \).

Example -- Line through two points

Given \( A(1, 2, -3) \) and \( B(4, -1, 5) \), \( \vec{d} = \begin{pmatrix} 3 \ -3 \ 8 \end{pmatrix} \). The line is \( \vec{r} = \begin{pmatrix} 1 \ 2 \ -3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \ -3 \ 8 \end{pmatrix} \).

Angles Between Vectors

• Use dot product: \( \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 ).
• Angle \( \theta \) satisfies \( \vec{a} \cdot \vec{b} = \lVert \vec{a} \rVert \lVert \vec{b} \rVert \cos\theta \).
• Perpendicular vectors have dot product zero; parallel vectors have dot product magnitude equal to product of lengths.

Example -- Angle calculation

Let \( \vec{a} = \begin{pmatrix} 2 \ -1 \ 3 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 1 \ 4 \ -2 \end{pmatrix} \).

\[ \vec{a} \cdot \vec{b} = 2 \times 1 + (-1) \times 4 + 3 \times (-2) = -8. \]

\[ \lVert \vec{a} \rVert = \sqrt{14}, \quad \lVert \vec{b} \rVert = \sqrt{21}, \quad \cos\theta = \frac{-8}{\sqrt{14} \sqrt{21}}. \]

Planar Geometry Basics

• Coplanar points share the same plane; check by showing direction vectors are linearly dependent.
• Area of triangle from two vectors \( \vec{u}, \vec{v} \): \( \tfrac{1}{2} \lVert \vec{u} \times \vec{v} \rVert \).
• Scalar triple product \( [\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) \) gives volume of parallelepiped; zero implies coplanarity.

Calculator Workflow

• Store vectors in GC matrices or vector memory for quick dot products and norms.
• Use RREF to test linear dependence and solve vector equations.
• Document cross product commands if your GC supports them; otherwise show manual determinant evaluation.

Exam Watch Points

• Express final answers as exact radicals unless context demands decimals.
• Always quote the parameter range when interpreting line equations (e.g. segment \( 0 \leq \lambda \leq 1 \)).
• Draw diagrams for vector proofs—markers award method marks for clear geometry.
• Distinguish between position vectors and direction vectors to avoid mixing origins.

Quick Revision Checklist

• [ ] Convert between vector, parametric, and component descriptions fluently.
• [ ] Calculate magnitudes, unit vectors, and angles with precise working.
• [ ] Determine lines through points and analyse parallelism/perpendicularity correctly.
• [ ] Apply scalar triple product tests for coplanarity and volumes.

Next steps: Advance to Topic 3.2 for dot/cross product applications in projections and geometry.