If A = 1i + 1j + 1k and B = 1i + 2j + 3k, find A · B.

If A = 1i + 1j + 1k and B = 1i + 2j + 3k, find A · B.

Step 1: Identify the components of vector A. A = 1i + 1j + 1k means A has components (1, 1, 1).
Step 2: Identify the components of vector B. B = 1i + 2j + 3k means B has components (1, 2, 3).
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 components into the formula: A · B = (1 * 1) + (1 * 2) + (1 * 3).
Step 5: Calculate each multiplication: (1 * 1) = 1, (1 * 2) = 2, (1 * 3) = 3.
Step 6: Add the results together: 1 + 2 + 3 = 6.

Dot Product – The dot product of two vectors is calculated by multiplying their corresponding components and summing the results.
Vector Notation – Understanding the representation of vectors in terms of their components along the i, j, and k axes.