Given vectors A = i + 2j + 3k and B = 4i + 5j + 6k, calculate A · B.

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Question: Given vectors A = i + 2j + 3k and B = 4i + 5j + 6k, calculate A · B.

Options:

Correct Answer: 30

Solution:

A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.

Given vectors A = i + 2j + 3k and B = 4i + 5j + 6k, calculate A · B.

Given vectors A = i + 2j + 3k and B = 4i + 5j + 6k, calculate A · B.

Step 1: Identify the components of vector A. A = i + 2j + 3k means A has components: A_x = 1, A_y = 2, A_z = 3.
Step 2: Identify the components of vector B. B = 4i + 5j + 6k means B has components: B_x = 4, B_y = 5, 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 = (1 * 4) + (2 * 5) + (3 * 6).
Step 5: Calculate each multiplication: 1 * 4 = 4, 2 * 5 = 10, 3 * 6 = 18.
Step 6: Add the results of the multiplications together: 4 + 10 + 18 = 32.

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