If A = i + 2j + 3k and B = 4i + 5j + 6k, what is the value of A · B?

If A = i + 2j + 3k and B = 4i + 5j + 6k, what is the value of 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: Multiply the corresponding components of A and B. For i: 1 * 4, for j: 2 * 5, and for k: 3 * 6.
Step 4: Calculate the products: 1 * 4 = 4, 2 * 5 = 10, and 3 * 6 = 18.
Step 5: Add the results of the products together: 4 + 10 + 18.
Step 6: Calculate the final sum: 4 + 10 = 14, then 14 + 18 = 32.
Step 7: The value of A · B is 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.