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

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

Step 1: Identify the components of vector A. A = i + 2j + 3k means A has components: 1 (for i), 2 (for j), and 3 (for k).
Step 2: Identify the components of vector B. B = 4i + 5j + 6k means B has components: 4 (for i), 5 (for j), and 6 (for k).
Step 3: Use the formula for the dot product A · B, which is calculated as (A's i component * B's i component) + (A's j component * B's j component) + (A's k component * B's k component).
Step 4: Substitute the values into the formula: (1 * 4) + (2 * 5) + (3 * 6).
Step 5: Calculate each multiplication: 1 * 4 = 4, 2 * 5 = 10, and 3 * 6 = 18.
Step 6: Add the results of the multiplications together: 4 + 10 + 18.
Step 7: The final result is 32, so A · B = 32.

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