Hi, this is the grade 12 calculus and vectors subject. This is the unit 7 vectors unit. I would really appreciate it if someone could answer the following questions with easy to follow steps and explanations. The questions are done by another student already, but I'm struggling to understand. So i would really appreciate the help for me to better be prepared for my test coming up and i'd be sure to include a good review.

Round all magnitudes to one decimal place, unless stated otherwise, and any angle measures to the nearest degree. 12. Determine the projection of rons given r = [8, 7, 4], s = [5, 8, -9]. Write answer in simplified form. [3] ( 8) (5) + (7) (8 ) + ( 4 ) ( - 9 ) [5,8,- 9] 151 = 1170 752 + 83 ( - 9) 2 = 13.0 60 So 131 =170 13.0 ( CS ,8, - 9]) 300 480 - 540 13 73' 13 geo or cartesian vectors components : magnitude & direction n airplane is flying at an airspeed of 404 kmuh on a heading of 0750. The wind is blowing at 29 km/h from a bearing of 143 404 km/h 0750 29 KM/h 143 b) Draw a diagram of the airspeed's vector, as a Cartesian vector. Write its components as exact values only. [1] Draw a diagram of the wind's vector. as a Cartesian vector. Write its components as exact values only. [2] ( ) Hint: Remember that a Cartesian vector's tail is Determine the components of the resultant vector (representing the velocity of the airplane). Draw a diagram of this vector and write its compone d) Determine the m ponents, as both exact values and approximate values rounded to 3 decimal places. [2] magnitude and ction of the resultant vector. Write your final answer (the velocity of the airplane) using appropriate units and the direction as a true bearing. (#) 13 a ) Q - 404 KM/ " C) R = atw = [ 404 cos 15 + 29 cos37, 404 sin 15 a= (404 cos 15 , 404 sin 15] + 29 sin 37] R = [ 413 . 394 , 122. 015] b ) Nix - I d) IR'1= 1413.394+122. 015? w= 29 km/L = 431. 025 KM/L - w = [29 cos (37, 29 sil37 ] on a bearing of OF4 X tane = 413:394 122.015 9= 74' 14. Given parallelogram ABCD, with vertices A(-5.-5), B(1, 3). C(8. 1), and D(2. -7), use vectors to determine the interior angles of the parallelogram [6) F = BC position vectors = [-5-1 ,-5-3] = [8-1, 1-3] e' =[.6, -8]- 7:[7,-27 Cose = e'. f* le'llFT Cose =- 26 9 : 94 2 D = 180 94" : Angles are 940, 86, 94, 860 = 86 in the parallelogram .10. For the following geometric vectors, determine axb . State whether your resultant vector will point into or out of the page. (3] axb = (all5 /sing = (16) (10 ) sin 117 in 1 al = 16 : 142. 6 n, where in 117- Is a unit vector pointing out of the page. 161 = 10 16. Given vectors m = [8. -28, 24] and n = [-2, 7, -6], show two ways to prove that the vectors are collinear. Explain your answers. 131 4 If Kn' . m m and it are collinear then the vectors are collinear If the angle between them IS 180 : Please - 4n= m -356 cose = (4 19 ) (89) redo and write stepss :. M and n are collinear 0 = 180 explain because they are multiples 180 of eachother : They are allinear . 17. Given vector d = [2, 26, -16], determine a unit vector in the same direction as vector a, and a unit vector in the opposite direction as vector a. Write answers as exact values, in simplified form. [3] U = unit vector w= a Please redo and write steps / = [2,26,-16] U= 3Va6 3126 ' 3 Val 13 explain Va" +267 ( -16) ? - u = 3 Va6 3 V 26' 306 2 ) - 13 [2,26,-16] 6V2618. Given vectors v = [9, -1, 5] and w = [4. -3, -7], determine the area of the parallelogram defined by vectors and w. [3] Sunits ? A = 1 v x wl This was correct but I don't understand ... Please write steps = 17-(-15), 20-1-63),-27-(-4)]1 and an explanation to help me get the 1[ 22, 83 , - 23]1 right answer myself! = V222+837 (-23)? = 37878 A = 88. 9 units? 19. Determine the volume of the parallelepiped defined by a = [7. -2, 6], b = [2, 1, -1], and c = [3, 0, -3]. [3]units V= 1a. ( 5 x c')1 = 1[7,-2, 6] . [-3-0,-3-(-6), 0-371 = 1(7,-2, 6] . [-3, 3,- 371 = 1(7X-3)+(-2)(3)+ (6) (-3)1 Please redo with steps! = 1-451 explanation Seperate question : What would you do V = 45 units if It was perpendicular