Example 1: Determine la I in the example above. Example 3: Find values of xand ysuch that 1,*) * ftr) = f t'] [+J-[v]-[-zJ

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1 A measurement which only describes the magnitude (i.e. size) of the object is called a scalar quantity, e.g. Glasgow is 11 miles from Airdrie. A vector is a quantity with magnitude and direction, e.g. Glasgow is 11 mifes from Airdrie on a bearing of 270". The line joining the origin to the point A (3, 4) can be described as: 13) t4) Comoonents ' Note that if OA = a, then AO = - The magnitude of a vector is its length, which can be determined by Pythagoras'Theorem. The magnitude of 4 is written as la l. r d,, -. lf u= 1 _ l, then lg = 1a'+b' \Dl Example 1: Determine la I in the example above. Two (or more) vectors can be added together to produce a resultant vector. For example, we could thinkof 4abovebeingthe resultantvectorof theaddition 13) r0) [o/ t 4] of x= l_ land y= -1. In general: AE+ nc=nc Example 2t If p + g = 7, findf when (2\ (7\ e= uno n= [-rj l_r). Example 3: Find values of xand ysuch that 1,*) * ftr) = f t'] [+J-[v]-[-zJ Subtraction of vectors can be considered as going along the second vector in the opposite direction. Page 57 of 82

2 If we go along 4 twice, the resultant vector is A + e - 2. As we have not changedirection, it follows that 2a must be parallel to g. Examole 4=tf b= f o ) and c= 1.- 2l a) 3p [f;,0'0, b) 2b+ e oe-lo A unit vector is a vector with magnitude = 1. Example 5: Find the components of the unit vector parallel to f = (i-) Consider the vector AB in tn" diagram opposite. AB ir tn" resultant vector of going along 4 in the opposite direction, followed by b in the correct direction. So, m = -g+ b,i.e.: Example 6: A is the point (4,2), B is the point (-5,-3). Find the componens of AB. Page 58 of 82

3 We have seen collinearity the Straight Line chapter; this can also be proven using vectors. Example 7: Prove the points F (6, 1), G (4,4), and H (-2, 13) are collinear, and find the ratio in which G divides FF. P divides AB in the ratio 2:3. By examining the diagram, we can find a formula for p (i.e. OP ). 0F=oT+AF In general, if P divides AB in the ratio m:n, then: Example 8: A is the point (3, 2) and B is the point (7, 14). Find the coordinates of P such that P divides AB in the ratio 1:3. Page 59 of 82

4 The position of a point in 3-D space can be described if we add a third coordinate to indicate height. Example 9: OABC DEFG is a cuboid, where F is the F (5, 4, 3) point (5, 4, 3). Write down the coordinates of the points: a)a b)d c)g d) M, the centre of face ABFE All rules for 2D to 3D In the diagram opposite, Vectors in 3D can also be described in terms of the three unit vectors i = which are parallel to the x, y, and zaxes respectively. ExampleLO=u=3! +2j -6k a) Express y in component form b) Find ly I The scalar product is the result of a type of multiplication of two vectors to give a scalar quantity (i.e. a number with no directional component). For vectors A and!, the scalar product (or dot product) is given as: Note:. a and! point away from the veftex o Q<0<1800 Page 60 of 82

5 Example 11: Find the scalar product in each case below, where I Al = 6 and I p1 = a) b) c) 10. We can use this formula to find the scalar product when we have been given the component forms of the two vectors but not the angle between them. Example 12: A is the point (L,2,3), B (6, 5, 4), and C (-!, -2,-6). Evaluate AE.Bd. A special case of the scalar product occurs when we have perpendicular vectors, i.e. when 0 = 90o. In the case opposite, a.b = lallbl cos9oo = lalldl x 0 =Q Example 13: P, Q, and R are the points (I, L, 2), (-L,-1,0) and (3,-4,-L) respectively. Find the components of QF and QR, and hence show that the vectors are perpendicular. Page 61 of 82

6 The first version of the scalar product includes the angle between the vectors, so we can rearrange this formula to obtain: Example 14= g = i+ 2j + 2k, and b= 2!+ 3j '6t. Find the angle between aand b. Forvectors d, b, andg:,a Exampfe 15: la = 5 and 12 = 8. Find a.(a+ b) Past Paper Example 1r The diagram shows a square-based pyramid of height 8 units. Square OABC has a side length of 6 units. The coordinates of A and D are (6, 0, 0) and (3, 3, 8). C lies on the y- axis. a) Write down the coordinates of B. b) Determine the components of DIand DB. Page 52 of 82

7 c) Calculate the size of ZADB. Past Paper Example 2: PQRSTU is a regular hexagon of side 2 units. P8, QF, and E represent the vectors e, b, and g: respectively. Find the value of a.@- e) Vectors: Topic Checklist Topic Ouestions Done Helo? Maqnitude of a vector c Exercise 13N, p 247, Q 2 Y/N v/n Resultant vectors c Exercise 13N, p 248, Q 5, 6,9, II Y/N Y/N Multiplication bv a scalar c Exercise 13N, p 248, Q 10, 11 Y/N Y/N Unit Vectors c Exercise 13F, p 238, Q I,2 Y/N Y/N Position Vectors and Exercise 13G, p 239, Q L,2 Y/N Y/N c Components Exercise 13N, p 249, O 13, L4 Y/N Y/N c Exercise 13N. p 249, O 15 Y/N Y/N Collinearity AIB Exercise 13N. o 249, O Y/N Y/N c Exercise 13N, p 249, Section Formula Q 20 Y/N Y/N A/B Exercise 13N, p 249, Q 2L,22,24 Y/N Y/N Scalar Product (anqle) c Exercise 13O, p 250, Q 1 Y/N Y/N Scalar Product (component) c Exercise 13P, p 252, Q L,2 Y/N Y/N Angle c Exercise 13O, p 253, between vectors O L,2 Y/N Y/N AIB Exercise 13S. o 255. O 4-7 Y/N Y/N Perpendicular c Exercise 13R. o 254, O L,2,3,5 Y/N Y/N Vectors NE Exercise 13R, p 254, Q 4, 6 - I Y/N Y/N Propefties of Scalar Product c Exercise 13U, p 257, Q 1, 2, 4, 5 Y/N Y/N Page 63 of 82