vectors.ms (1989B)
1 .TL 2 Vectors in comp sci 3 .AU 4 Lucas Standen 5 .AI 6 QMC 7 .2C 8 9 .EQ 10 delim @@ 11 .EN 12 13 .EQ 14 delim @# 15 .EN 16 17 .NH 1 18 How to write them 19 20 .LP 21 To write a vector, like in maths we can use 22 .EQ 23 ({i sub x, j sub y}) 24 .EN 25 But they can also be written 26 .EQ 27 R sup 2 28 .EN 29 30 .EQ 31 R sup 3 32 .EN 33 Where the power is the number of degrees available 34 35 .NH 1 36 Combining vectors 37 .LP 38 To combine vectors one can use the formula 39 .EQ 40 w = alpha u + beta v 41 .EN 42 Where w is the combined vector and 43 .EQ 44 alpha + beta = 1 45 .EN 46 47 .NH 2 48 Example 49 .EQ 50 u = (2,2) 51 .EN 52 53 .EQ 54 v = (6,-2) 55 .EN 56 57 We can then say that 58 .EQ 59 w = (4, 0) 60 .EN 61 By subtracting v from u 62 63 Then using the formula 64 .EQ 65 2 alpha + 6 beta = 3 66 .EN 67 Where 3 is a point on the combined vector 68 .EQ 69 2 alpha + -2 beta = 1 70 .EN 71 72 We can then solve for @ beta # like so 73 74 .EQ 75 6 beta - 3 = -2 beta - 1 76 .EN 77 78 .EQ 79 8 beta - 2 = 0 80 .EN 81 82 .EQ 83 8 beta = 2 84 .EN 85 86 .EQ 87 beta = 2 over 8 88 .EN 89 90 .EQ 91 beta = 1 over 4 92 .EN 93 94 From this we can say 95 .EQ 96 alpha = 3 over 4 97 .EN 98 Because 99 .EQ 100 alpha + beta = 1 101 .EN 102 103 .NH 2 104 Another example 105 106 .EQ 107 2 alpha + 6 beta = 2 108 .EN 109 110 .EQ 111 2 alpha - 2 beta = 1 112 .EN 113 114 .EQ 115 8 beta = 1 116 .EN 117 118 .EQ 119 beta = 1 over 8 120 .EN 121 122 .EQ 123 2 alpha - 2 ({1 over 8}) = 1 124 .EN 125 126 .EQ 127 2 alpha = 5 over 4 128 .EN 129 130 .EQ 131 alpha = 5 over 8 132 .EN 133 134 Since 135 .EQ 136 alpha + beta != 1 137 .EN 138 We can say that w does not lie on the vector uv 139 140 And because it is greater than 1 it means it is inside the triangle created by u and v 141 142 .NH 1 143 The dot product 144 145 .LP 146 To solve use the following formula 147 148 .EQ 149 u.v = |u|.|v| cos( theta ) 150 .EN 151 152 Where @ theta # is the angle between the 2 vectors and 153 154 .EQ 155 |u| = " magnitude of u, " sqrt {x sup 2 + y sup 2} 156 .EN 157 158 You can also use 159 .EQ 160 u.v = u sub 1 . v sub 1 + u sub 2 . v sub 2 + u sub n + v sub n ... 161 .EN 162 If you don't have the angle 163 164 Don't be confused by the dot, it just means 165 .EQ 166 u sub 1 . v sub 1 = u sub 1 times v sub 1 167 .EN 168 169 .NH 2 170 Exam question 171 172 .LP 173 1.1) 174 .EQ 175 |b| = 4 176 .EN 177 178 1.2) 179 .EQ 180 u.v = u sub 1 . v sub 1 + u sub 2 . v sub 2 + u sub n + v sub n ... 181 .EN 182 183 .EQ 184 a.b = 4 . 4 + 3 . 0 185 .EN 186 187 .EQ 188 a.b = 16 + 0 189 .EN 190 191 .EQ 192 a.b = 16 193 .EN