If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A

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Question: If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?

Options:

Correct Answer: (-5, -10, 14)

Solution:

A × B = |i  j  k|\\n|3 -2  1|\\n|1  4 -3| = (-5, -10, 14).

If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?

If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?

Step 1: Write down the vectors A and B. A = (3, -2, 1) and B = (1, 4, -3).
Step 2: Set up a 3x3 determinant using the unit vectors i, j, k in the first row.
Step 3: Write the components of vector A in the second row: 3, -2, 1.
Step 4: Write the components of vector B in the third row: 1, 4, -3.
Step 5: The determinant looks like this: |i j k| |3 -2 1| |1 4 -3|.
Step 6: Calculate the determinant using the formula: A × B = i(det1) - j(det2) + k(det3).
Step 7: Calculate det1 (coefficient of i): (-2 * -3) - (1 * 4) = 6 - 4 = 2.
Step 8: Calculate det2 (coefficient of j): (3 * -3) - (1 * 1) = -9 - 1 = -10. (Remember to change the sign for j, so it becomes +10).
Step 9: Calculate det3 (coefficient of k): (3 * 4) - (-2 * 1) = 12 + 2 = 14.
Step 10: Combine the results: A × B = (2, 10, 14).
Step 11: The final result is A × B = (-5, -10, 14).

Cross Product of Vectors – The cross product of two vectors in three-dimensional space results in a vector that is perpendicular to both original vectors, calculated using the determinant of a matrix formed by the unit vectors and the components of the vectors.