cross_product.md (7530B)
1 # Cross Product 2 3 The **cross product** is an operation that takes two vectors in 4 $$ \mathbb{R}^3 $$ 5 6 and returns another vector in $$ \mathbb{R}^3 $$ 7 8 , written $\mathbf{a} \times \mathbf{b}$. Unlike the dot product, the result is 9 a vector, not a scalar. The cross product is only defined in three dimensions 10 (and in a generalized sense in seven dimensions; here we restrict to 11 $\mathbb{R}^3$). 12 13 ## Geometric meaning 14 15 - **Direction:** $\mathbf{a} \times \mathbf{b}$ is perpendicular to both 16 $\mathbf{a}$ and $\mathbf{b}$, following the right-hand rule: if you point 17 your fingers along $\mathbf{a}$ and curl them toward $\mathbf{b}$, your thumb 18 points in the direction of $\mathbf{a} \times \mathbf{b}$. 19 - **Magnitude:** $\|\mathbf{a} \times \mathbf{b}\| = 20 \|\mathbf{a}\|\,\|\mathbf{b}\|\sin\theta$, where $\theta$ is the angle between 21 $\mathbf{a}$ and $\mathbf{b}$. So the length equals the area of the 22 parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$. 23 24 ## Algebraic definition 25 26 For vectors 27 28 $$ 29 \mathbf{a} = 30 \begin{pmatrix} 31 a_1\\ 32 a_2\\ 33 a_3 34 \end{pmatrix}, 35 \quad 36 \mathbf{b} = 37 \begin{pmatrix} 38 b_1\\ 39 b_2\\ 40 b_3 41 \end{pmatrix}, 42 $$ 43 44 the cross product is 45 46 $$ 47 \mathbf{a} \times \mathbf{b} = 48 \begin{pmatrix} 49 a_2 b_3 - a_3 b_2\\ 50 a_3 b_1 - a_1 b_3\\ 51 a_1 b_2 - a_2 b_1 52 \end{pmatrix}. 53 $$ 54 55 This can be remembered using the determinant of a formal $3\times 3$ matrix: 56 57 $$ 58 59 \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & 60 \mathbf{e}_3\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{vmatrix} 61 62 $$ 63 64 $$ 65 66 \mathbf{e}_1(a_2 b_3 - a_3 b_2) 67 68 - \mathbf{e}_2(a_1 b_3 - a_3 b_1) 69 70 * \mathbf{e}_3(a_1 b_2 - a_2 b_1), 71 72 $$ 73 74 where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard unit vectors 75 in $\mathbb{R}^3$. 76 77 ### The "Cross-Out" Method (Fastest) 78 79 The shorthand calculation for this is: 80 81 1. Stack them: Write the components of the first vector over the second vector 82 twice. 83 2. Cross out the first and last columns. 84 3. Multiply in an 'X' pattern (top-left bot-right minus top-right bot-left) for 85 each remaining pair: 86 87 <img src="/assets/cross_product_shorthand.png" alt="Cross Product Calculation" width="500"> 88 89 ## Rules of calculation (with examples in LaTeX) 90 91 Let $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^3$ and $\lambda \in 92 \mathbb{R}$. 93 94 --- 95 96 **1. Anticommutativity** 97 98 Swapping the order flips the sign: 99 100 $$ 101 \mathbf{a} \times \mathbf{b} = -\bigl(\mathbf{b} \times \mathbf{a}\bigr). 102 $$ 103 104 Example: 105 106 $$ 107 \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} \times \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix} 108 = \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix}, 109 \quad 110 \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix} \times \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} 111 = \begin{pmatrix} 0\\ 0\\ -1 \end{pmatrix}. 112 $$ 113 114 --- 115 116 **2. Distributivity over addition** 117 118 $$ 119 \mathbf{a} \times (\mathbf{b} + \mathbf{c}) 120 = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}, 121 \qquad 122 (\mathbf{a} + \mathbf{b}) \times \mathbf{c} 123 = \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c}. 124 $$ 125 126 Example (second component of $\mathbf{a} \times (\mathbf{b}+\mathbf{c})$): 127 128 $$ 129 \mathbf{a} = \begin{pmatrix} 1\\ 2\\ 0 \end{pmatrix},\; 130 \mathbf{b} = \begin{pmatrix} 0\\ 1\\ 1 \end{pmatrix},\; 131 \mathbf{c} = \begin{pmatrix} 1\\ 0\\ 1 \end{pmatrix} 132 \;\Rightarrow\; 133 \mathbf{b}+\mathbf{c} = \begin{pmatrix} 1\\ 1\\ 2 \end{pmatrix}. 134 $$ 135 136 $$ 137 \mathbf{a} \times \mathbf{b} = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix},\quad 138 \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 2\\ -1\\ -2 \end{pmatrix} 139 \;\Rightarrow\; 140 \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4\\ -2\\ -1 \end{pmatrix}. 141 $$ 142 143 $$ 144 \mathbf{a} \times (\mathbf{b}+\mathbf{c}) = \begin{pmatrix} 2\cdot 2 - 0\cdot 1\\ 0\cdot 1 - 1\cdot 2\\ 1\cdot 1 - 2\cdot 1 \end{pmatrix} = \begin{pmatrix} 4\\ -2\\ -1 \end{pmatrix}. 145 $$ 146 147 --- 148 149 **3. Scalar multiplication (homogeneity)** 150 151 A scalar can be factored out of either slot: 152 153 $$ 154 (\lambda \mathbf{a}) \times \mathbf{b} 155 = \mathbf{a} \times (\lambda \mathbf{b}) 156 = \lambda (\mathbf{a} \times \mathbf{b}). 157 $$ 158 159 Example: with $\mathbf{a} = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}$, 160 $\mathbf{b} = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix}$, $\lambda = 3$, 161 162 $$ 163 (3\mathbf{a}) \times \mathbf{b} 164 = \begin{pmatrix} 3\\ 0\\ 0 \end{pmatrix} \times \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix} 165 = \begin{pmatrix} 0\\ 0\\ 3 \end{pmatrix} 166 = 3 \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix} 167 = 3(\mathbf{a} \times \mathbf{b}). 168 $$ 169 170 --- 171 172 **4. Cross product with the zero vector** 173 174 $$ 175 \mathbf{a} \times \mathbf{0} = \mathbf{0} \times \mathbf{a} = \mathbf{0}. 176 $$ 177 178 --- 179 180 **5. Parallel vectors** 181 182 $\mathbf{a}$ and $\mathbf{b}$ are parallel (or one is zero) if and only if 183 184 $$ 185 \mathbf{a} \times \mathbf{b} = \mathbf{0}. 186 $$ 187 188 Example: $\mathbf{a} = \begin{pmatrix} 2\\ 4\\ 6 \end{pmatrix}$, $\mathbf{b} 189 = \begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix} = \tfrac{1}{2}\mathbf{a}$, so 190 191 $$ 192 \mathbf{a} \times \mathbf{b} 193 = \begin{pmatrix} 4\cdot 3 - 6\cdot 2\\ 6\cdot 1 - 2\cdot 3\\ 2\cdot 2 - 4\cdot 1 \end{pmatrix} 194 = \begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}. 195 $$ 196 197 --- 198 199 **6. Self-cross product** 200 201 $$ 202 \mathbf{a} \times \mathbf{a} = \mathbf{0}. 203 $$ 204 205 (Special case of the parallel-vectors rule.) 206 207 --- 208 209 **7. Jacobi identity** 210 211 $$ 212 213 \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) 214 215 - \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) 216 - \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0}. 217 218 $$ 219 220 --- 221 222 **8. Relation to dot product (vector triple product expansion)** 223 224 $$ 225 \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) 226 = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}. 227 $$ 228 229 Example: $\mathbf{a} = \mathbf{e}_1$, $\mathbf{b} = \mathbf{e}_2$, 230 $\mathbf{c} = \mathbf{e}_3$: 231 232 $$ 233 \mathbf{a} \cdot \mathbf{c} = 0,\quad \mathbf{a} \cdot \mathbf{b} = 0 234 \;\Rightarrow\; 235 \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = 0\cdot \mathbf{b} - 0\cdot \mathbf{c} = \mathbf{0}. 236 $$ 237 238 $$ 239 \mathbf{b} \times \mathbf{c} = \mathbf{e}_1 240 \;\Rightarrow\; 241 \mathbf{e}_1 \times \mathbf{e}_1 = \mathbf{0}. 242 $$ 243 244 --- 245 246 **9. Magnitude and angle** 247 248 $$ 249 \|\mathbf{a} \times \mathbf{b}\|^2 250 = \|\mathbf{a}\|^2 \|\mathbf{b}\|^2 - (\mathbf{a} \cdot \mathbf{b})^2. 251 $$ 252 253 Equivalently, $\|\mathbf{a} \times \mathbf{b}\| = 254 \|\mathbf{a}\|\,\|\mathbf{b}\|\sin\theta$. 255 256 --- 257 258 **10. Relation to the dot product (scalar triple product)** 259 260 $$ 261 \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) 262 = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) 263 = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}). 264 $$ 265 266 This value is the (signed) volume of the parallelepiped spanned by $\mathbf{a}, 267 \mathbf{b}, \mathbf{c}$. Example: 268 269 $$ 270 \mathbf{a} = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix},\; 271 \mathbf{b} = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix},\; 272 \mathbf{c} = \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix} 273 \;\Rightarrow\; 274 \mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix},\quad 275 \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 1. 276 $$ 277 278 ## Worked example 279 280 Compute $\mathbf{u} \times \mathbf{v}$ for 281 282 $$ 283 \mathbf{u} = \begin{pmatrix} 2\\ -1\\ 3 \end{pmatrix},\qquad 284 \mathbf{v} = \begin{pmatrix} 1\\ 4\\ -2 \end{pmatrix}. 285 $$ 286 287 $$ 288 \mathbf{u} \times \mathbf{v} 289 = \begin{pmatrix} 290 (-1)(-2) - (3)(4)\\ 291 (3)(1) - (2)(-2)\\ 292 (2)(4) - (-1)(1) 293 \end{pmatrix} 294 = \begin{pmatrix} 295 2 - 12\\ 296 3 + 4\\ 297 8 + 1 298 \end{pmatrix} 299 = \begin{pmatrix} -10\\ 7\\ 9 \end{pmatrix}. 300 $$ 301 302 Check: $\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = 2(-10) + (-1)(7) + 303 3(9) = -20 - 7 + 27 = 0$, and $\mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) 304 = 1(-10) + 4(7) + (-2)(9) = -10 + 28 - 18 = 0$, so the result is perpendicular 305 to both $\mathbf{u}$ and $\mathbf{v}$.