If A = 4i - 3j and B = -2i + 5j, calculate A · B.

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Question: If A = 4i - 3j and B = -2i + 5j, calculate A · B.

Options:

Correct Answer: -7

Solution:

A · B = (4)(-2) + (-3)(5) = -8 - 15 = -23.

If A = 4i - 3j and B = -2i + 5j, calculate A · B.

If A = 4i - 3j and B = -2i + 5j, calculate A · B.

Step 1: Identify the components of vector A. A = 4i - 3j means A has a component of 4 in the i direction and -3 in the j direction.
Step 2: Identify the components of vector B. B = -2i + 5j means B has a component of -2 in the i direction and 5 in the j direction.
Step 3: Use the formula for the dot product A · B, which is A · B = (A_i * B_i) + (A_j * B_j).
Step 4: Substitute the components into the formula. A_i = 4, B_i = -2, A_j = -3, B_j = 5.
Step 5: Calculate the product of the i components: (4) * (-2) = -8.
Step 6: Calculate the product of the j components: (-3) * (5) = -15.
Step 7: Add the results from Step 5 and Step 6: -8 + (-15) = -8 - 15 = -23.
Step 8: The final result of A · B is -23.

Dot Product of Vectors – The dot product of two vectors A and B is calculated by multiplying their corresponding components and summing the results.