Given vectors A = (2, -1, 3) and B = (4, 0, -2), find A × B.

Given vectors A = (2, -1, 3) and B = (4, 0, -2), find A × B.

Correct Answer: (-1, -10, 4)

Step 1: Write down the vectors A and B. A = (2, -1, 3) and B = (4, 0, -2).
Step 2: Set up the determinant for the cross product using the unit vectors i, j, k.
Step 3: Create a 3x3 matrix with the first row as the unit vectors (i, j, k), the second row as the components of vector A, and the third row as the components of vector B.
Step 4: The matrix looks like this: | i j k |
Step 5: | 2 -1 3 |
Step 6: | 4 0 -2 |
Step 7: Calculate the determinant using the formula for the cross product.
Step 8: For the i component, calculate: (-1 * -2) - (3 * 0) = 2.
Step 9: For the j component, calculate: -(2 * -2 - 3 * 4) = -(-4 - 12) = 16.
Step 10: For the k component, calculate: (2 * 0) - (-1 * 4) = 0 + 4 = 4.
Step 11: Combine the components to get the result: A × B = (2, 16, 4).

Vector Cross Product – The cross product of two vectors in three-dimensional space results in a vector that is perpendicular to both original vectors.
Determinants – The calculation of the cross product involves finding the determinant of a 3x3 matrix formed by the unit vectors and the components of the two vectors.