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

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

Options:

Correct Answer: -1

Solution:

A · B = (5)(3) + (-2)(4) + (1)(-2) = 15 - 8 - 2 = 5.

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

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

Step 1: Identify the components of vector A. A = 5i - 2j + k means A has components: A_x = 5, A_y = -2, A_z = 1.
Step 2: Identify the components of vector B. B = 3i + 4j - 2k means B has components: B_x = 3, B_y = 4, B_z = -2.
Step 3: Use the formula for the dot product A · B, which is A · B = (A_x * B_x) + (A_y * B_y) + (A_z * B_z).
Step 4: Substitute the values into the formula: A · B = (5 * 3) + (-2 * 4) + (1 * -2).
Step 5: Calculate each multiplication: 5 * 3 = 15, -2 * 4 = -8, and 1 * -2 = -2.
Step 6: Add the results together: 15 + (-8) + (-2) = 15 - 8 - 2.
Step 7: Perform the final calculation: 15 - 8 = 7, and then 7 - 2 = 5.
Step 8: The final result of A · B is 5.

Dot Product – The dot product of two vectors is calculated by multiplying their corresponding components and summing the results.
Vector Components – Understanding how to identify and use the components of vectors in calculations.