What is the cross product of vectors E = i + 2j and F = 3i + 4j?

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Question: What is the cross product of vectors E = i + 2j and F = 3i + 4j?

Options:

Correct Answer: -2k

Solution:

E × F = |i  j  k| |1  2  0| |3  4  0| = (0 - 0)i - (0 - 0)j + (1*4 - 2*3)k = -2k.

What is the cross product of vectors E = i + 2j and F = 3i + 4j?

What is the cross product of vectors E = i + 2j and F = 3i + 4j?

Step 1: Identify the vectors E and F. E = i + 2j can be written as (1, 2, 0) and F = 3i + 4j can be written as (3, 4, 0).
Step 2: Set up the determinant for the cross product using the unit vectors i, j, k and the components of E and F.
Step 3: Write the determinant as a 3x3 matrix: |i j k| |1 2 0| |3 4 0|.
Step 4: Calculate the determinant. The formula for the determinant is: i*(2*0 - 0*4) - j*(1*0 - 0*3) + k*(1*4 - 2*3).
Step 5: Simplify each part: The i component is (0 - 0)i = 0i, the j component is (0 - 0)j = 0j, and the k component is (4 - 6)k = -2k.
Step 6: Combine the results: E × F = 0i + 0j - 2k = -2k.

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