If A = 5i + 2j + 3k and B = 1i - 4j + 2k, find A · B.

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Question: If A = 5i + 2j + 3k and B = 1i - 4j + 2k, find A · B.

Options:

Correct Answer: 0

Solution:

A · B = (5)(1) + (2)(-4) + (3)(2) = 5 - 8 + 6 = 3.

If A = 5i + 2j + 3k and B = 1i - 4j + 2k, find A · B.

If A = 5i + 2j + 3k and B = 1i - 4j + 2k, find A · B.

Step 1: Identify the components of vector A, which are 5i, 2j, and 3k.
Step 2: Identify the components of vector B, which are 1i, -4j, and 2k.
Step 3: Multiply the corresponding components of A and B: (5 * 1) for i, (2 * -4) for j, and (3 * 2) for k.
Step 4: Calculate the products: 5 * 1 = 5, 2 * -4 = -8, and 3 * 2 = 6.
Step 5: Add the results of the products together: 5 + (-8) + 6.
Step 6: Simplify the addition: 5 - 8 = -3, then -3 + 6 = 3.
Step 7: The final result of A · B is 3.

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, j, and k components is essential for performing vector operations.