ch46 Vectors by Chtan FYKulai Chapter 46 Vectors ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai A VECTOR? Describes the motion of an object A Vector comprises Direction Magnitude We will consider Column Vectors General Vectors Vector Geometry Size ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Column Vectors NOTE! Label is in BOLD. When handwritten, draw a wavy line under the label i.e. Vector a a 2 up 4 RIGHT COLUMN Vector ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Column Vectors Vector b b 2 up 3 LEFT COLUMN Vector? ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Column Vectors Vector u n 2 down 4 LEFT COLUMN Vector? ch46 Vectors by Chtan FYKulai

Describe these vectors a b c d ch46 Vectors by Chtan FYKulai

Alternative labelling F B D E G C A H ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai General Vectors A Vector has BOTH a Length & a Direction All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k k k k k k can be in any position ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai General Vectors B Line CD is Parallel to AB k CD is TWICE length of AB D A 2k Line EF is Parallel to AB E C EF is equal in length to AB -k EF is opposite direction to AB F ch46 Vectors by Chtan FYKulai

Write these Vectors in terms of k B D 2k F G 1½k ½k E C -2k A H ch46 Vectors by Chtan FYKulai

Combining Column Vectors k A B C D ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Simple combinations A B C ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Vector Geometry Consider this parallelogram Q O P R a b Opposite sides are Parallel OQ is known as the resultant of a and b ch46 Vectors by Chtan FYKulai

Resultant of Two Vectors Is the same, no matter which route is followed Use this to find vectors in geometrical figures ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.1 S is the Midpoint of PQ. Work out the vector Q O P R a b . S = a + ½b ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Alternatively S is the Midpoint of PQ. Work out the vector Q O P R a b . S = b + a - ½b = ½b + a = a + ½b ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.2 A B C p q M AC= p, AB = q M is the Midpoint of BC Find BC BC BA AC = + = -q + p = p - q ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.3 A B C p q M AC= p, AB = q M is the Midpoint of BC Find BM BM ½BC = = ½(p – q) ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.4 A B C p q M AC= p, AB = q M is the Midpoint of BC Find AM AM + ½BC = AB = q + ½(p – q) = q +½p - ½q = ½q +½p = ½(q + p) = ½(p + q) ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Alternatively A B C p q M AC= p, AB = q M is the Midpoint of BC Find AM AM + ½CB = AC = p + ½(q – p) = p +½q - ½p = ½p +½q = ½(p + q) ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Distribution’s law : The scalar multiplication of a vector : 𝑘 𝒂+𝒃 =𝑘𝒂+𝑘𝒃 𝑘 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑘>0 𝑜𝑟 𝑘<0 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Other important facts : ℎ𝑘 𝒂 = ℎ𝑘 𝒂 ℎ+𝑘 𝒂=ℎ𝒂+𝑘𝒂 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai A vector with the starting point from the origin point is called position vector. 位置向量 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Every vector can be expressed in terms of position vector. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.5 Given that 𝒂= 2 5 , 𝒃= −2 3 and also 𝑘𝒂+𝑙𝒃= 10 1 . Find the values of 𝑘 𝑎𝑛𝑑 𝑙. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.6 Given that 𝒂=𝑚𝑖−4𝑗, 𝒃=3𝑖−2𝑗, and 𝒂 𝒂𝒏𝒅 𝒃 are parallel. Find the value of m. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.7 𝐴𝐵 = 3 −2 , 𝐵𝐶 = 2 5 , a point 𝐶 1,4 . Find the coordinates of 𝐴 𝑎𝑛𝑑 𝐵, then express point 𝐶 in terms of 𝒊 𝑎𝑛𝑑 𝒋 . ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.8 If 𝑃 3,5 , 𝑃𝑄 = 5 −7 , find the coordinates of 𝑄. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.9 Given that 𝒂=2𝑖+𝑝𝑗, 𝒃= 7+𝑝 𝑖+4𝑗, and 𝒂 𝒂𝒏𝒅 𝒃 are parallel. Find the value of 𝑝. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Magnitude of a vector 𝐴 𝑖𝑠 𝑥 1 , 𝑦 1 , 𝐵 𝑖𝑠 𝑥 2 , 𝑦 2 . 𝒂 =𝑨𝑩 = 𝒙 𝟐 − 𝒙 𝟏 𝟐 + 𝒚 𝟐 − 𝒚 𝟏 𝟐 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai 𝒙,𝒚 𝒂 𝒂 = 𝒙 𝟐 + 𝒚 𝟐 𝑦 𝑥 Unit vector : 𝒂 = 𝟏 𝒂 ∙𝒂 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.10 Find the magnitude of the vectors : 𝒂 𝒑= −𝟐 𝟓 (b) 𝒓=𝟗𝒊−𝟏𝟐𝒋 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.11 Find the unit vectors in e.g. 10 : 𝒂 𝒑= −𝟐 𝟓 (b) 𝒓=𝟗𝒊−𝟏𝟐𝒋 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Ratio theorem 𝒚 A P B 𝒙 𝟎 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.12 M is the midpoint of AB, find in terms of . ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.13 2 𝑷 3 P divides AB into 2:3. Find in terms of . 𝑨 𝑩 𝑶 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Application of vector in plane geometry e.g.14 In the diagram, CB=4CN, NA=5NX, M is the midpoint of AB. A M X B C N (a) Express the following vectors in terms of u and v ; (i) (ii) ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai (b) Show that (c) Calculate the value of (i) (ii) ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Soln: (a) (i) (ii) (b) ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai (c) (i) (ii) ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.15 A M and N are midpoints of AB, AC. Prove that N M C B ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.16 B In the diagram K divides AD into 1:l, and divides BC into 1:k . 2a 1 A 1 K 6a l k D O C 2b 6b Express position vector OK in 2 formats. Find the values of k and l. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai More exercises on this topic : 高级数学高二下册 Pg 33 Ex10g ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Scalar product of two vectors If a and b are two non-zero vectors, θ is the angle between the vectors. Then , ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Scalar product of vectors satisfying : Commutative law : Associative law : Distributive law : ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.17 Find the scalar product of the following 2 vectors : ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.18 If , find the angle between them. If are perpendicular, find k. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Scalar product (special cases) 1. Two perpendicular vectors Unit vector for y-axis N.B. Unit vector for x-axis ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai 2. Two parallel vectors N.B. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.19 Given , Find . Ans:[17/2] ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Scalar product (dot product) The dot product can also be defined as the sum of the products of the components of each vector as : ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.20 Given that Find (a) (b) angle between a and b . Ans: (a) 25 (b) 45° ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Applications of Scalar product 高级数学高二下册 Pg 42 to pg43 Eg30 to eg 33 ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai More exercises on this topic : 高级数学高二下册 Pg 44 Ex10i Misc 10 ch46 Vectors by Chtan FYKulai

Miscellaneous Examples ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai e.g.21 Given that D, E, F are three midpoints of BC, CA, AB of a triangle ABC. Prove that AD, BE and CF are concurrent at a point G and . ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Soln: A F E G From ratio theorem B D C ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai We select a point G on AD such that 𝑨𝑮 𝑮𝑫 =𝟐. From ratio theorem, Similarly, We select a G1 point on BE such that 𝑩 𝑮 𝟏 𝑮 𝟏 𝑬 =𝟐. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Similarly, We select a G2 point on CF such that 𝑪 𝑮 𝟐 𝑮 𝟐 𝑭 =𝟐. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Because g1, g2, g are the same, G, G1, G2 are the same point G! G is on AD, BE and CF, hence AD, BE and CF intersect at G. And also 𝑨𝑮 𝑮𝑫 = 𝑩𝑮 𝑮𝑬 = 𝑪𝑮 𝑮𝑭 =𝟐 is established. ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai Centroid of a ∆ ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai

ch46 Vectors by Chtan FYKulai The end ch46 Vectors by Chtan FYKulai