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

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

Options:

Correct Answer: 1k

Solution:

A × B = |i  j  k| |3  4  0| |4  3  0| = (0 - 0)i - (0 - 0)j + (9 - 12)k = -3k.

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

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

Step 1: Identify the components of vector A and vector B. Vector A = 3i + 4j means A has components (3, 4, 0) and vector B = 4i + 3j means B has components (4, 3, 0).
Step 2: Set up the determinant for the cross product using the unit vectors i, j, k and the components of A and B. This looks like: |i j k| |3 4 0| |4 3 0|.
Step 3: Calculate the determinant. The formula for the determinant is: i*(4*0 - 0*3) - j*(3*0 - 0*4) + k*(3*3 - 4*4).
Step 4: Simplify each part. The i component is (4*0 - 0*3) = 0, the j component is (3*0 - 0*4) = 0, and the k component is (3*3 - 4*4) = 9 - 12 = -3.
Step 5: Combine the results. The cross product A × B = 0i - 0j - 3k, which simplifies to -3k.

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 the components of vectors in the i, j, k format is essential for performing vector operations.