What is the cross product of vectors A = i + 2j + 3k and B = 4i + 5j + 6k?

What is the cross product of vectors A = i + 2j + 3k and B = 4i + 5j + 6k?

Step 1: Write down the vectors A and B. A = i + 2j + 3k and B = 4i + 5j + 6k.
Step 2: Set up a 3x3 determinant using the unit vectors i, j, k in the first row, the components of vector A in the second row, and the components of vector B in the third row.
Step 3: The determinant looks like this: | i j k |
Step 4: Fill in the second row with the components of vector A: | 1 2 3 |
Step 5: Fill in the third row with the components of vector B: | 4 5 6 |
Step 6: Now calculate the determinant using the formula for a 3x3 matrix: A × B = i(2*6 - 3*5) - j(1*6 - 3*4) + k(1*5 - 2*4).
Step 7: Calculate each part: i(12 - 15) = -3i, -j(6 - 12) = 6j, k(5 - 8) = -3k.
Step 8: Combine the results: A × B = -3i + 6j - 3k.

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