If A = 5i - 2j and B = -3i + 4j, what is the value of A · B?

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Question: If A = 5i - 2j and B = -3i + 4j, what is the value of A · B?

Options:

Correct Answer: -23

Solution:

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

If A = 5i - 2j and B = -3i + 4j, what is the value of A · B?

If A = 5i - 2j and B = -3i + 4j, what is the value of A · B?

Step 1: Identify the components of vector A. A = 5i - 2j means A has a component of 5 in the i direction and -2 in the j direction.
Step 2: Identify the components of vector B. B = -3i + 4j means B has a component of -3 in the i direction and 4 in the j direction.
Step 3: Use the formula for the dot product A · B, which is A · B = (A's i component) * (B's i component) + (A's j component) * (B's j component).
Step 4: Substitute the values into the formula: A · B = (5) * (-3) + (-2) * (4).
Step 5: Calculate the first part: (5) * (-3) = -15.
Step 6: Calculate the second part: (-2) * (4) = -8.
Step 7: Add the results from Step 5 and Step 6: -15 + (-8) = -15 - 8.
Step 8: Final calculation: -15 - 8 = -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.