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

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

Options:

Correct Answer: 25

Solution:

A · B = (5)(5) + (5)(-5) = 25 - 25 = 0.

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

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

Step 1: Identify the vectors A and B. A = 5i + 5j and B = 5i - 5j.
Step 2: Write down the formula for the dot product of two vectors. The dot product A · B = (A_x * B_x) + (A_y * B_y), where A_x and A_y are the components of vector A, and B_x and B_y are the components of vector B.
Step 3: Identify the components of A and B. For A, A_x = 5 and A_y = 5. For B, B_x = 5 and B_y = -5.
Step 4: Substitute the components into the dot product formula: A · B = (5 * 5) + (5 * -5).
Step 5: Calculate the first part: 5 * 5 = 25.
Step 6: Calculate the second part: 5 * -5 = -25.
Step 7: Add the results from Step 5 and Step 6: 25 + (-25) = 25 - 25.
Step 8: Simplify the result: 25 - 25 = 0.

Dot Product – The dot product of two vectors A and B 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 essential for calculating the dot product.