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

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

Options:

Correct Answer: 24

Solution:

A · B = (1)(2) + (2)(3) + (3)(4) = 2 + 6 + 12 = 20.

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

Given vectors A = i + 2j + 3k and B = 2i + 3j + 4k, 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 = 2i + 3j + 4k means B has components: B_x = 2, B_y = 3, B_z = 4.
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 * 2) + (2 * 3) + (3 * 4).
Step 5: Calculate each multiplication: (1 * 2) = 2, (2 * 3) = 6, (3 * 4) = 12.
Step 6: Add the results together: 2 + 6 + 12 = 20.

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