S56 (5.3) Vectors.notebook January 29, 2016


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1 Dily Prctice Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution of the ltitude from in the tringle (7, 3), (1, 6)nd C( 5, 1) Vectors vector is quntity with oth length nd direction. Revision: Vectors (Components nd Mgnitude) Write the components nd mgnitude for ech of the following: D (i) (ii) Vectors re equl if they hve the sme length (mgnitude) & direction. C D C Vector Informtion The negtive vector is vector with the sme mgnitude ut goes in the opposite direction. The sign of ech component is reversed. re vectors tht originte from the origin. = = Unit Vectors re vectors with mgnitude 1. E.g. O is clled the zero vector nd is written 0
2 The vector cn lso e found y using the position vectors. Write down the vector (5, 3) = (5, 3) (2, 1) = (2, 1) Q1. Dily Prctice Tody we will e lerning out position vectors & collinerity. PSP Trgets 1. Given P(1, 2) nd Q(3, 4), wht is the vector? Q 2. Find the components of when X(1, 2) nd Y(3, 4) P
3 Collinerity Collinerity using vectors Points re colliner if they re on the sme line. C To prove tht 2 points re colliner, we could show tht they shre common point & find the grdients nd show tht they re equl. Thinking out the vectors nd, how cn we show tht, nd C re colliner? Collinerity using vectors C The vectors will e prllel (one will e multiple/frction of the other) e.g. nd they will shre common point i.e.. Tody we will e proving points re colliner using vectors. Collinerity 1. Prove tht the following points re colliner (3, 4), (1, 8) nd C(0, 10) Dily Prctice Q1. Given f(x) =. Find n expression for h(x) where h(x) = f(f(x)), giving your nswer s frction in its simplest form. Q2. Show tht the line with eqution y = 2x + 1 does not intersect the prol with eqution y = x 2 + 3x + 4
4 S56 (5.3) Vectors.noteook Jnury 29, 2016 Collinerity 2. Tody we will e continuing to lern out collinerity. Vectors (i) HW Online due Dily Prctice Q1. Solve 3sin x + 5sinx  2 = 0 where 0 x Q2. The digrm shows circle, centre C(0, 3) with tngent drwn t the point P(2, 0). Stte the eqution of this tngent Tody we will e lerning how to divide line in given rtio. P(2, 0) (0, 3) Homework due Tuesdy. Dividing line in rtio dvice: Drw digrm of the given line nd mrk in the rtio. Note the vector of the line segment Work ckwrds to find the point. 1. H divides PQ in the rtio 1:3. Find the coordintes of H if P(3, 2) nd Q(7, 14) 2. P divides in the rtio 3:2 where is the point (3, 2, 6) nd is the point (7, 3, 1). Wht re the coordintes of P?
5 Q1. Dily Prctice Q2. Tody we will e writing 3Dimensionl vectors in component form nd lerning out the Sclr Product. Homework Due Tuesdy. 3D Vectors in Component Form 3D Vectors in Component Form vector my e defined in terms of i, j nd k where i, j nd k re unit vectors in the directions of ech of the xes. Questions: 1. Clculte the mgnitude of r when r = 2i  3j + 4k In component form, these re written s 2. u = 3i  4j + 2k nd v = i + 5j  k, if u = v, find. Sometimes vector my e expressed s comintion of its components. Eg. If v = 3i + 4j + k then Sclr Product (Or Dot Product) Sclr Product (Or Dot Product) oth vectors need to e pointing towrds the ngle or wy from the ngle for. = cosθ. Sclr product is the multipliction of vectors to get sclr quntity. θ θ Formul given in exm pper: Mgnitude For the cses elow. =  cosθ θ θ Component Form If nd re perpendiculr, then. = 0 ecuse cos90 0 = 0
6 Dily Prctice Q1. Express 3x 26x + 11 in the form r(x + p) 2 + q Q2. sequence is defined y the recurrence reltion U n+1 = 0.5U n + 1, U 0 = 4, (i) Wht is the vlue of U 2? (ii) Find the limit of this recurrence reltion Tody we will e continuing to lern out Sclr Product. Homework Online: Due tomorrow Sclr Product (Or Dot Product) 1. Clculte. for the pir of vectors shown when = 4 nd = Dily Prctice Q1. Stte the eqution of the line prllel to 3x  y + 2 = 0 tht psses through (2, 3) Q2. Stte the eqution of the line shown in the digrm 2. Clculte the sclr product for the pir of vectors Q3. Find f'(x) when f(x) = 3x 2  x 1/ Exercise 13 O Q1. () (c) (d) (f) Ex. 13P Q1. (), (e) Q2. () Q3 () Q4. Given two vectors in component form, how would we find the ngle inetween? Tody we will e lerning how to find the ngle etween 2 vectors.
7 Sclr Product (Or Dot Product) Finding the ngle To find the ngle etween pir of vectors, just rerrnge the formul. Sclr Product (Or Dot Product) Finding the ngle 1. Clculte the ngle etween the pir of vectors Sclr Product (Or Dot Product) Finding the ngle 2. Show tht nd re perpendiculr Dily Prctice Q1. Functions f(x) = nd g(x) = 2x + 3 re defined on suitle domin. Find n expression for h(x) where h(x) = g(f(x)) Ex. 13Q Q1. () (d) Q2. Q4. Ex. 13R Q1. Q3. Q5. Q7. Q2. Vectors u nd v re defined y u = 3i + 2j + 0k nd v = 2i  3j + 4k. Determine whether they re perpendiculr to ech other. Properties of Sclr Product Some of the properties of the sclr product: Tody we will e lerning out some properties of the Sclr Product. Homework Online due x.x = x 2 x.y = y.x rckets cn e expnded e.g. x(x + y) = x.x + x.y
8 Using Properties of the Sclr Product Exmple:
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