If the vectors A = 2i + 2j and B = 2i - 2j, find A · B.

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Question: If the vectors A = 2i + 2j and B = 2i - 2j, find A · B.

Options:

Correct Answer: 4

Solution:

A · B = (2)(2) + (2)(-2) = 4 - 4 = 0.

If the vectors A = 2i + 2j and B = 2i - 2j, find A · B.

If the vectors A = 2i + 2j and B = 2i - 2j, find A · B.

Step 1: Identify the components of vector A. A = 2i + 2j means A has a component of 2 in the i direction and 2 in the j direction.
Step 2: Identify the components of vector B. B = 2i - 2j means B has a component of 2 in the i direction and -2 in the j direction.
Step 3: Use the formula for the dot product of two vectors. The dot product A · B is calculated as (A_i * B_i) + (A_j * B_j).
Step 4: Substitute the components into the formula. A_i = 2, B_i = 2, A_j = 2, B_j = -2.
Step 5: Calculate the first part of the dot product: (2)(2) = 4.
Step 6: Calculate the second part of the dot product: (2)(-2) = -4.
Step 7: Add the results from Step 5 and Step 6: 4 + (-4) = 0.
Step 8: Conclude that the dot product A · B = 0.

Dot Product – The dot product of two vectors is calculated by multiplying their corresponding components and summing the results.
Vector Components – Understanding how to break down vectors into their i and j components is crucial for performing operations like the dot product.