Vectors in simple words
A vector is like an arrow. It has a size (how long it is) and a direction. We write it as a list of parts, such as (1, 2, 3), which means 1 step along x, 2 along y and 3 along z.
To add two vectors, add the matching parts: (1, 2, 3) + (4, 5, 6) = (5, 7, 9).
The dot product multiplies the matching parts and adds them: 1 × 4 + 2 × 5 + 3 × 6 = 4 + 10 + 18 = 32. A dot product of 0 means the arrows are at right angles.
The length (magnitude) comes from Pythagoras: square each part, add them and take the square root. For (1, 2, 3) that is √(1 + 4 + 9) = √14 ≈ 3.74.
Formula and method
The dot product sums corresponding component products. For nonzero vectors, cos θ = (a·b)/(|a||b|).
Worked example
Vectors (1, 0, 0) and (0, 1, 0) have dot product 0, angle 90° and cross product (0, 0, 1).
Worked example with substitution
For a = (1, 2, 3) and b = (4, 5, 6): a · b = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32. The magnitudes are |a| = √(1² + 2² + 3²) = √14 ≈ 3.7417 and |b| = √77 ≈ 8.7750.
The angle is θ = arccos[32 ÷ (3.7417 × 8.7750)] = arccos(0.97463) ≈ 12.93°. The cross product is (2×6 − 3×5, 3×4 − 1×6, 1×5 − 2×4) = (−3, 6, −3).