[H2 MATH] Chapter 5 - Vectors

SHALYN TAY COPYRIGHTED © A LEVEL H2 MATHEMATICS VECTORS

CHAPTER ANALYSIS MASTERY EXAM WEIGHTAGE SHALYN TAY COPYRIGHTED ©

BASIC VECTOR PROPERTIES VECTOR ALGEBRA PARALLEL AND NON - PARALLEL VECTORS SCALAR PRODUCT (DOT PRODUCT) VECTOR PRODUCT (CROSS PRODUCT) VECTORS I SHALYN TAY COPYRIGHTED ©

Basic Vector Properties SHALYN TAY COPYRIGHTED © • A scalar quantity has magnitude but no associated direction (e.g. distance and speed). • A vector quantity has both magnitude and direction (e.g. displacement and velocity). • A position vector defines the position of a point relative to another. • A free/displacement vector is a vector with no associated position. Modulus of Vector = Magnitude or Length of Vector If 𝒂 = 𝒙 𝒚 𝒛 then 𝑎 = 𝑥2 + 𝑦2 + 𝑧2 𝑶 is the origin, from which we usually associate our position vectors. 𝑶𝑨 = 𝒂 = 𝟒 𝟕 𝟏𝟏 = 𝟒𝒊 + 𝟕𝒋 + 𝟏𝟏𝒌 𝑶𝑨 = 𝒂 = 42 + 72 + 112 𝒊, 𝒋 𝒂𝒏𝒅 𝒌 are unit vectors. z y x O 𝒊 = 𝟏 𝟎 𝟎 𝒋 = 𝟎 𝟏 𝟎 𝒌 = 𝟎 𝟎 𝟏 A (4, 7, 11) 𝒂 𝑶 = 𝟎 𝟎 𝟎 A unit vector, denoted by ෝ𝒂, is a vector whose magnitude is 1. ෝ𝒂 = 𝒂 𝒂 The zero/null vector is a vector with zero magnitude and no direction. 𝟎 = 𝟎 𝟎 𝟎

SHALYN TAY COPYRIGHTED © The negative of a vector has the same magnitude as a vector but is opposite in direction (i.e. a and –a). 𝒂 𝒂 −𝒃𝒃 𝒂 + 𝒃 𝒂 − 𝒃 O A B C 𝒂 + 𝒃 = 𝑶𝑨 + 𝑶𝑩 = 𝑶𝑨 + 𝑨𝑪 = 𝑶𝑩 + 𝑩𝑪 𝒂 − 𝒃 = 𝒂 + −𝒃 = 𝑩𝑶 + 𝑶𝑨 = 𝑶𝑨 − 𝑶𝑩 = 𝑩𝑪 + 𝑪𝑨 = 𝑩𝑨 In general, 𝑼𝑽 = 𝑶𝑽 − 𝑶𝑼 𝒂 ± 𝒃 = 𝒙𝟏 𝒚𝟏 𝒛𝟏 ± 𝒙𝟐 𝒚𝟐 𝒛𝟐 = 𝒙𝟏 ± 𝒙𝟐 𝒚𝟏 ± 𝒚𝟐 𝒛𝟏 ± 𝒛𝟐 Vector Algebra If 𝑨𝑩 = 𝑶𝑩 − 𝑶𝑨 = 𝒙𝟏 − 𝒙𝟐 𝒚𝟏 − 𝒚𝟐 𝒛𝟏 − 𝒛𝟐 𝑨𝑩 = (𝒙𝟏 − 𝒙𝟐)2+(𝒚𝟏−𝒚𝟐)2 + (𝒛𝟏−𝒛𝟐)2 Equal Vectors Vectors are equal when they have the same direction and magnitude. If 𝑨𝑩 = 𝒅 𝒆 𝒇 and 𝑪𝑫 = 𝒑 𝒒 𝒓 , 𝑨𝑩 = 𝑪𝑫, then 𝒅 𝒆 𝒇 = 𝒑 𝒒 𝒓 and d = p, e = q, and f = r Scalar Multiplication When vector 𝒂 is multiplied by the scalar 𝜆, the magnitude of the vector changes and the vector 𝜆𝒂 has magnitude 𝜆 times of 𝒂 (i.e. 𝝀𝒂 = 𝝀 𝒂 ) • If 𝜆 > 0, 𝜆𝒂 and 𝒂 are in the same direction • If 𝜆 = 0, 𝜆𝒂 is a zero vector i.e. 𝜆𝒂 = 𝟎 • If 𝜆 < 0, 𝜆𝒂 and 𝒂 are in opposite directions Laws of Vector Algebra 1. 𝒂 + 𝒃 = 𝒃 + 𝒂 2. 𝜆𝒂 = 𝒂𝜆 3. 𝒂 + 𝒃 + 𝒄 = 𝒂 + 𝒃 + 𝒄 4. (𝜆𝜇)𝒂 = 𝜆(𝜇𝒂) 5. 𝜆 + 𝜇 𝒂 = 𝜆𝒂 + 𝜇𝒂 6. 𝜆 𝒂 + 𝒃 = 𝜆𝒂 + 𝜆𝒃

SHALYN TAY COPYRIGHTED © Parallel & Non-Parallel Vectors 𝒂 𝒃 We say 𝒂 is parallel to 𝒃 if and only if 𝒃 = 𝜆𝒂 for some 𝜆 ∈ ℝ \ {0} 𝑃 𝑄 𝑅 Collinearity Three points, P , Q and R are collinear if and only if 𝑷𝑸 ∕∕ 𝑷𝑹 , with P as the common point i.e. 𝑷𝑸 = 𝜆𝑷𝑹 for some 𝜆 ∈ ℝ \ {0} ෝ𝒂 A unit vector, denoted by ෝ𝒂, is a vector whose magnitude is 1. ෝ𝒂 = 𝒂 𝒂 𝒂 𝒓 𝒃 𝜆 𝜇 O The Ratio Theorem 𝐴 𝐵 𝑅 To find 𝑶𝑹 = 𝒓 : 𝒓 = 𝜇𝒂 + 𝜆𝒃 𝜆 + 𝜇If R is the midpoint of AB, then R divides AB in the ratio 1:1 and 𝑶𝑹 = 𝒓 = 1 2 (𝒂 + 𝒃)

SHALYN TAY COPYRIGHTED © Scalar Product (Dot Product) 𝜃 𝜃 Acute Obtuse The scalar product of two vectors a and b, is defined as 𝒂 ∙ 𝒃 = 𝒂 𝒃 cos 𝜃 𝜃 is the angle between a and b such that a and b are either both leaving from or both meeting at the same point 𝒂 ∙ 𝒃 > 0 𝒂 ∙ 𝒃 < 0 𝒂 ∙ 𝒃 is called a scalar product because the product is a scalar 𝒂 ∙ 𝒃 = 𝒙𝟏 𝒚𝟏 𝒛𝟏 ∙ 𝒙𝟐 𝒚𝟐 𝒛𝟐 = 𝒙𝟏 𝒙𝟐 + 𝒚𝟏𝒚𝟐 + 𝒛𝟏𝒛𝟐 Scalar Product Properties 1. 𝒂 ∙ 𝒃 = 𝒃 ∙ 𝒂 2. 𝒂 ∙ 𝒃 ± 𝒄 = (𝒂 ∙ 𝒃) ± (𝒂 ∙ 𝒄) 3. 𝝀 𝒂 ∙ 𝒃 = λ𝒂 ∙ 𝒃 = 𝒂 ∙ (𝝀𝒃) 4. 𝒂 ∙ 𝒂 = 𝒂 𝟐 Angle Between Two Non-Zero Vectors cos 𝜃 = 𝒂 ∙ 𝒃 𝒂 𝒃 Length of Projection 𝑶𝑷 = 𝒂 ∙ ෡𝒃 = 𝒂 ∙ 𝒃 𝒃 𝐴 𝜃 𝒂 𝒃 𝐵 𝑂 𝑃 Perpendicular Vectors Two non-zero vectors a and b are perpendicular i.e. 𝒂 ⊥ 𝒃 , if and only if 𝒂 ∙ 𝒃 = 𝟎 Link this to direction cosines, which is the cosine of the angle between a vector and the x- , y- and z-axes

SHALYN TAY COPYRIGHTED © Vector Product (Cross Product) The vector product of two vectors a and b, is defined as 𝒂 × 𝒃 = ( 𝒂 𝒃 sin 𝜃)ෝ𝒏 𝜃 is the angle between a and b, and ෝ𝒏 is the unit vector perpendicular to both a and b (unit vector of normal) 𝒂 × 𝒃 is called a vector product because the product is a vector 𝒂 × 𝒃 = 𝒙𝟏 𝒚𝟏 𝒛𝟏 × 𝒙𝟐 𝒚𝟐 𝒛𝟐 = −( 𝒚𝟏𝒛𝟐 − 𝒛𝟏𝒚𝟐 𝒙𝟏𝒛𝟐 − 𝒛𝟏𝒙𝟐) 𝒙𝟏𝒚𝟐 − 𝒚𝟏𝒙𝟐 Vector Product Properties 1. 𝒂 × 𝒃 = −(𝒃 × 𝒂) 2. 𝒂 × 𝒃 ± 𝒄 = (𝒂 × 𝒃) ± (𝒂 × 𝒄) 3. 𝝀 𝒂 × 𝒃 = λ𝒂 × 𝒃 = 𝒂 × (𝝀𝒃) 4. 𝒂 × 𝒂 = 𝟎 5. 𝒂 × 𝒃 = 𝒃 × 𝒂 = 𝒂 𝒃 sin 𝜃 Parallel Vectors Two non-zero vectors a and b are parallel if and only if 𝒂 × 𝒃 = 𝟎 B A O 𝒂 𝒃 𝜽 ෝ𝒏 𝒂 𝒃 O A B C Area of Triangle OAB = 𝟏 𝟐 𝒂 × 𝒃 Area of Parallelogram OACB = 𝒂 × 𝒃

EQUATIONS OF STRAIGHT LINES CALCULATIONS FOR A POINT AND A LINE CALCULATIONS FOR A PAIR OF LINES VECTORS II SHALYN TAY COPYRIGHTED ©

SHALYN TAY COPYRIGHTED © Equations of Straight Lines Vector Equation 𝒂 𝒃 𝒓 O 𝑙𝑅 𝐴 𝑙 ∶ 𝒓 = 𝒂 + 𝜆𝒃 , 𝜆 ∈ ℝ Any point on the line Direction vector of line Parametric Equation ቐ 𝑥 = 𝒂𝟏 + 𝜆𝒃𝟏 𝑦 = 𝒂𝟐 + 𝜆𝒃𝟐 𝑧 = 𝒂𝟑 + 𝜆𝒃𝟑 , 𝜆 ∈ ℝ From vector equation, let r = 𝒙 𝒚 𝒛 to transform into parametric equation Cartesian Equation 𝜆 = 𝒙 − 𝒂𝟏 𝒃𝟏 𝜆 = 𝒚 − 𝒂𝟐 𝒃𝟐 𝜆 = 𝒛 − 𝒂𝟑 𝒃𝟑 , 𝜆 ∈ ℝMake 𝜆 the subject 𝒙 − 𝒂𝟏 𝒃𝟏 = 𝒚 − 𝒂𝟐 𝒃𝟐 = 𝒛 − 𝒂𝟑 𝒃𝟑 , 𝜆 ∈ ℝ Equate 𝜆 𝑥 = 𝒂𝟏 , 𝒚 − 𝒂𝟐 𝒃𝟐 = 𝒛 − 𝒂𝟑 𝒃𝟑 If 𝒃𝟏 = 𝟎, cartesian equations becomes