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Q: What does H2 Maths Notes (JC 1-2): 3.1) Basic Properties of Vectors cover?A: Vector notation, magnitude-direction forms, and line representations for H2 Maths Topic 3.1.Before you revise Core Definitions A vector
v ⃗ \vec{v} v has magnitude
∥ v ⃗ ∥ \lVert \vec{v} \rVert ∥ v ∥ Part 12 of 32
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and direction; in component form
v ⃗ = ( v 1 v 2 v 3 ) \vec{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix} v = ( v 1 v 2 v 3 ) .
Position vector of point
A ( x , y , z ) A(x, y, z) A ( x , y , z ) is
O A ⃗ = ( x y z ) \vec{OA} = \begin{pmatrix} x \ y \ z \end{pmatrix} O A = ( x y z ) .
Magnitude:
∥ v ⃗ ∥ = v 1 2 + v 2 2 + v 3 2 \lVert \vec{v} \rVert = \sqrt{v_1^2 + v_2^2 + v_3^2} ∥ v ∥ = v 1 2 + v 2 2 + v 3 2 .
Unit vector:
v ^ = v ⃗ ∥ v ⃗ ∥ \hat{v} = \dfrac{\vec{v}}{\lVert \vec{v} \rVert} v ^ = ∥ v ∥ v .
Vector Operations Addition/subtraction component-wise:
a ⃗ ± b ⃗ = ( a 1 ± b 1 a 2 ± b 2 a 3 ± b 3 ) \vec{a} \pm \vec{b} = \begin{pmatrix} a_1 \pm b_1 \ a_2 \pm b_2 \ a_3 \pm b_3 \end{pmatrix} a ± b = ( a 1 ± b 1 a 2 ± b 2 a 3 ± b 3 ) .
Scalar multiplication:
k a ⃗ = ( k a 1 k a 2 k a 3 ) k \vec{a} = \begin{pmatrix} ka_1 \ ka_2 \ ka_3 \end{pmatrix} k a = ( k a 1 k a 2 k a 3 ) Parallel vectors satisfy
a ⃗ = k b ⃗ \vec{a} = k \vec{b} a = k b .
Midpoint of segment
A B AB A B :
O M ⃗ = 1 2 ( O A ⃗ + O B ⃗ ) \vec{OM} = \tfrac{1}{2}(\vec{OA} + \vec{OB}) OM = 2 1 ( O A + OB ) If A ( 2 , − 1 , 5 ) A(2, -1, 5) A ( 2 , − 1 , 5 ) B ( − 4 , 7 , 1 ) B(-4, 7, 1) B ( − 4 , 7 , 1 )
O M ⃗ = 1 2 ( 2 + ( − 4 ) − 1 + 7 5 + 1 ) = ( − 1 3 3 ) .
\vec{OM} = \frac{1}{2}\begin{pmatrix} 2 + (-4) \ -1 + 7 \ 5 + 1 \end{pmatrix} = \begin{pmatrix} -1 \ 3 \ 3 \end{pmatrix}.
OM = 2 1 ( 2 + ( − 4 ) − 1 + 7 5 + 1 ) = ( − 1 3 3 ) .
Lines in Vector Form A line through point
A A A with direction vector
d ⃗ \vec{d} d is
r ⃗ = O A ⃗ + λ d ⃗ \vec{r} = \vec{OA} + \lambda \vec{d} r = O A + λ d .
Convert to parametric equations:
x = x A + λ d 1 x = x_A + \lambda d_1 x = x A + λ d 1 , etc.
Two points determine a line: direction vector
d ⃗ = O B ⃗ − O A ⃗ \vec{d} = \vec{OB} - \vec{OA} d = OB − O A Example -- Line through two points
Given A ( 1 , 2 , − 3 ) A(1, 2, -3) A ( 1 , 2 , − 3 ) B ( 4 , − 1 , 5 ) B(4, -1, 5) B ( 4 , − 1 , 5 ) d ⃗ = ( 3 − 3 8 ) \vec{d} = \begin{pmatrix} 3 \ -3 \ 8 \end{pmatrix} d = ( 3 − 3 8 ) r ⃗ = ( 1 2 − 3 ) + λ ( 3 − 3 8 ) \vec{r} = \begin{pmatrix} 1 \ 2 \ -3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \ -3 \ 8 \end{pmatrix} r = ( 1 2 − 3 ) + λ ( 3 − 3 8 )
Angles Between Vectors Example -- Angle calculation
Let a ⃗ = ( 2 − 1 3 ) \vec{a} = \begin{pmatrix} 2 \ -1 \ 3 \end{pmatrix} a = ( 2 − 1 3 ) b ⃗ = ( 1 4 − 2 ) \vec{b} = \begin{pmatrix} 1 \ 4 \ -2 \end{pmatrix} b = ( 1 4 − 2 )
a ⃗ ⋅ b ⃗ = 2 × 1 + ( − 1 ) × 4 + 3 × ( − 2 ) = − 8.
\vec{a} \cdot \vec{b} = 2 \times 1 + (-1) \times 4 + 3 \times (-2) = -8.
a ⋅ b = 2 × 1 + ( − 1 ) × 4 + 3 × ( − 2 ) = − 8.
∥ a ⃗ ∥ = 14 , ∥ b ⃗ ∥ = 21 , cos θ = − 8 14 21 .
\lVert \vec{a} \rVert = \sqrt{14}, \quad \lVert \vec{b} \rVert = \sqrt{21}, \quad \cos\theta = \frac{-8}{\sqrt{14} \sqrt{21}}.
∥ a ∥ = 14 , ∥ b ∥ = 21 , cos θ = 14 21 − 8 .
Planar Geometry Basics
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 ≤ λ ≤ 1 0 \leq \lambda \leq 1 0 ≤ λ ≤ 1 ).
Draw diagrams for vector proofs-markers award method marks for clear geometry.
Distinguish between position vectors and direction vectors to avoid mixing origins.
Practice Quiz Consolidate magnitude, direction, and coplanarity checks before tackling product operations.
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.
H2 Maths Notes (JC 1-2): 3.1) Basic Properties of Vectors
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gives volume of parallelepiped; zero implies coplanarity.
