If A = (2, 3, 4) and B = (1, 0, -1), find the vector product A × B.

If A = (2, 3, 4) and B = (1, 0, -1), find the vector product A × B.

Correct Answer: (3, 6, -3)

Step 1: Write down the vectors A and B. A = (2, 3, 4) and B = (1, 0, -1).
Step 2: Set up the determinant for the vector product A × B 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 (2, 3, 4), and the third row as the components of vector B (1, 0, -1).
Step 4: The matrix looks like this: | i j k |
Step 5: | 2 3 4 |
Step 6: | 1 0 -1 |
Step 7: Calculate the determinant of this matrix to find the vector product.
Step 8: Use the formula for the determinant: A × B = i(3 * -1 - 4 * 0) - j(2 * -1 - 4 * 1) + k(2 * 0 - 3 * 1).
Step 9: Simplify each component: i(-3) - j(-2 - 4) + k(0 - 3).
Step 10: This gives us: -3i + 6j - 3k.
Step 11: Write the final answer in vector form: A × B = (3, 6, -3).

Vector Product – The vector product (or 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 vector product involves the determinant of a 3x3 matrix formed by the unit vectors and the components of the two vectors.