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

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

Step 1: Identify the components of vector A. A = 2i + 3j means A has a component of 2 in the i direction and 3 in the j direction.
Step 2: Identify the components of vector B. B = 4i + k means B has a component of 4 in the i direction and 0 in the j direction (since there is no j term), and 1 in the k direction.
Step 3: Write down the components of A and B. A = (2, 3, 0) and B = (4, 0, 1).
Step 4: Use the dot product formula A · B = (A_i * B_i) + (A_j * B_j) + (A_k * B_k).
Step 5: Substitute the values into the formula: A · B = (2 * 4) + (3 * 0) + (0 * 1).
Step 6: Calculate each part: (2 * 4) = 8, (3 * 0) = 0, (0 * 1) = 0.
Step 7: Add the results together: 8 + 0 + 0 = 8.
Step 8: The final answer is A · B = 8.

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