If A = 5i + 2j + 3k and B = 4i - j + 6k, find A · B. (2020)

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Question: If A = 5i + 2j + 3k and B = 4i - j + 6k, find A · B. (2020)

Options:

Correct Answer: 32

Exam Year: 2020

Solution:

A · B = (5)(4) + (2)(-1) + (3)(6) = 20 - 2 + 18 = 36.

If A = 5i + 2j + 3k and B = 4i - j + 6k, find A · B. (2020)

If A = 5i + 2j + 3k and B = 4i - j + 6k, find A · B. (2020)

Step 1: Identify the components of vector A. A = 5i + 2j + 3k means A has components: A_x = 5, A_y = 2, A_z = 3.
Step 2: Identify the components of vector B. B = 4i - j + 6k means B has components: B_x = 4, B_y = -1, B_z = 6.
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 * 4) + (2 * -1) + (3 * 6).
Step 5: Calculate each multiplication: 5 * 4 = 20, 2 * -1 = -2, 3 * 6 = 18.
Step 6: Add the results together: 20 + (-2) + 18 = 20 - 2 + 18.
Step 7: Perform the final addition: 20 - 2 = 18, then 18 + 18 = 36.
Step 8: Conclude that A · B = 36.

Dot Product of Vectors – The dot product of two vectors A and B is calculated by multiplying their corresponding components and summing the results.