If vector A = 5i + 12j and vector B = 12i - 5j, what is the value of A × B?

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Question: If vector A = 5i + 12j and vector B = 12i - 5j, what is the value of A × B?

Options:

Correct Answer: -85

Solution:

A × B = |i  j  k| |5  12  0| |12 -5  0| = (0 - 0)i - (0 - 0)j + (5*-5 - 12*12)k = -85k.

If vector A = 5i + 12j and vector B = 12i - 5j, what is the value of A × B?

If vector A = 5i + 12j and vector B = 12i - 5j, what is the value of A × B?

Step 1: Identify the components of vector A and vector B. Vector A = 5i + 12j means A has components (5, 12, 0) and vector B = 12i - 5j means B has components (12, -5, 0).
Step 2: Set up the determinant for the cross product A × B using the unit vectors i, j, and k. This is done by creating a 3x3 matrix with the first row as the unit vectors, the second row as the components of vector A, and the third row as the components of vector B.
Step 3: Write the matrix: | i j k | | 5 12 0 | | 12 -5 0 |.
Step 4: Calculate the determinant of the matrix. The formula for the determinant is: A × B = i(0 - 0) - j(0 - 0) + k(5 * -5 - 12 * 12).
Step 5: Simplify the calculations: The i component is 0, the j component is 0, and the k component is (5 * -5) - (12 * 12) = -25 - 144 = -169.
Step 6: Therefore, the result of A × B is 0i + 0j - 169k, which simplifies to -169k.

Vector Cross Product – The cross product of two vectors in three-dimensional space is 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 essential for finding the cross product.
Vector Components – Recognizing the components of vectors in the i, j, k format is crucial for performing vector operations.