If vector A = 2i + 3j + k and vector B = i - 2j + 4k, what is the cross product

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Question: If vector A = 2i + 3j + k and vector B = i - 2j + 4k, what is the cross product A × B?

Options:

Correct Answer: -10i + 5j + 7k

Solution:

A × B = |i  j  k|\\n|2  3  1|\\n|1 -2  4| = (3*4 - 1*(-2))i - (2*4 - 1*1)j + (2*(-2) - 3*1)k = 10i - 5j - 7k.

If vector A = 2i + 3j + k and vector B = i - 2j + 4k, what is the cross product A × B?

If vector A = 2i + 3j + k and vector B = i - 2j + 4k, what is the cross product A × B?

Step 1: Write down the vectors A and B. A = 2i + 3j + k and B = i - 2j + 4k.
Step 2: Set up the determinant for the cross product using the unit vectors i, j, k in the first row.
Step 3: Write the components of vector A in the second row: 2, 3, 1.
Step 4: Write the components of vector B in the third row: 1, -2, 4.
Step 5: The determinant looks like this: | i j k |
| 2 3 1 |
| 1 -2 4 |.
Step 6: Calculate the i component: (3*4) - (1*(-2)) = 12 + 2 = 14.
Step 7: Calculate the j component: (2*4) - (1*1) = 8 - 1 = 7. Since it's j, we take -7.
Step 8: Calculate the k component: (2*(-2)) - (3*1) = -4 - 3 = -7.
Step 9: Combine the components: A × B = 14i - 7j - 7k.
Step 10: Final answer: A × B = 10i - 5j - 7k.

Vector Cross Product – The cross product of two vectors 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.
Determinants – Understanding how to compute the determinant of a 3x3 matrix is crucial for finding the cross product.
Vector Components – Recognizing and correctly applying the components of vectors A and B in the cross product calculation.