Given the following two vectors in R3:
calculate the cross product X × Y
X = 8i - 2j + 13k
Y = 3i + 5j - k
Write the coefficients as a matrix
| i | j | k |
| x1 | x2 | x3 |
| y1 | y2 | y3 |
| → |
| i | j | k |
| 8 | -2 | 13 |
| 3 | 5 | -1 |
Evaluate Cross Product Formula
X × Y = (x2y3 - x3y2)i + (x3y1 - x1y3)j + (x1y2 - x2y1)k
X × Y = (-2 x -1 - 13 x 5)i + (13 x 3 - 8 x -1)j + (8 x 5 - -2 x 3)k
X × Y = (2 - 65)i + (39 - -8)j + (40 - -6)k
X × Y = - 63i + 47j + 46k
Parallel Check:
Since X × Y <> 0:
X and Y are not parallel
Final Answer
X × Y = - 63i + 47j + 46k
Since X × Y <> 0:
X and Y are not parallel
Since X × Y <> 0:
X and Y are not parallel
