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

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

Options:

Correct Answer: -1

Solution:

A · B = (5)(3) + (2)(-4) = 15 - 8 = 7.

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

If A = 5i + 2j and B = 3i - 4j, find 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: Multiply the i components of A and B together. This is 5 (from A) times 3 (from B), which equals 15.
Step 4: Multiply the j components of A and B together. This is 2 (from A) times -4 (from B), which equals -8.
Step 5: Add the results from Step 3 and Step 4 together. This is 15 (from Step 3) plus -8 (from Step 4), which equals 7.
Step 6: The final result of A · B is 7.

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 (horizontal) and j (vertical) components is essential for performing operations like the dot product.