Dot Product

In coordinate form you multiply the matching coordinates and add the products: a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃. Example: a⃗ = (1|2|2) and b⃗ = (2|0|1) give 1·2 + 2·0 + 2·1 = 4. The result is a single number, not a vector; that is exactly where the name scalar (dot) product comes from. In two dimensions it works identically with two summands. There is also the geometric form a⃗·b⃗ = |a⃗|·|b⃗|·cos φ with the enclosed angle φ; both forms give the same value and are used depending on the given data. In exams you almost always need the coordinate form for computing and the angle form for interpreting the result.

Two nonzero vectors are perpendicular exactly when their dot product is zero. The proof is thus a one-line calculation: a⃗ = (2|1|−2) and b⃗ = (1|2|2) give 2·1 + 1·2 + (−2)·2 = 2 + 2 − 4 = 0, so a⃗ ⊥ b⃗. The reason sits in the angle form: a⃗·b⃗ = |a⃗|·|b⃗|·cos φ becomes zero exactly when cos φ = 0, that is φ = 90°. This criterion is the workhorse of analytic geometry: orthogonality of lines via their direction vectors, line-plane relationships via the normal vector, perpendicular heights and feet in triangle tasks. Conversely you construct perpendicular vectors by setting the dot product to zero and choosing one coordinate freely.

Rearrange the angle form for cos φ: cos φ = (a⃗·b⃗)/(|a⃗|·|b⃗|). So compute the dot product and both magnitudes, then apply arccos. Example: a⃗ = (1|2|2), b⃗ = (2|0|1): a⃗·b⃗ = 4, |a⃗| = 3, |b⃗| = √5, so cos φ = 4/(3√5) ≈ 0.596 and φ ≈ 53.4°. The sign of the dot product reveals the angle class in advance: positive means acute, zero means 90°, negative obtuse. Two cautions: if cos comes out beyond [−1; 1], you forgot to divide by the magnitudes. And for angles between lines you take the absolute value of the numerator, because direction vectors may be flipped and the intersection angle is at most 90°.

Because the dot product measures something different from a direction: it measures how strongly two vectors act in the same direction. Geometrically it projects b⃗ onto the direction of a⃗ and multiplies the projection length |b⃗|·cos φ by the length |a⃗|; the outcome is a measure of "aligned length times length", which is naturally a number. Physics makes this tangible: work is W = F⃗·s⃗, force component along the path times distance, a scalar without direction. Do not confuse it with the cross product, which really does yield a vector (perpendicular to both factors) and measures areas. Mnemonic: dot product → scalar (number) → angles and lengths; cross product → vector → normals and areas.

The dot product of a vector with itself gives its squared magnitude: a⃗·a⃗ = a₁² + a₂² + a₃² = |a⃗|². So the length is |a⃗| = √(a⃗·a⃗), and this is exactly the Pythagorean theorem in space: the squares of the coordinates add up to the square of the length. Example: a⃗ = (1|2|2) has |a⃗| = √(1 + 4 + 4) = 3. The law of cosines is inside too: |a⃗ − b⃗|² = |a⃗|² + |b⃗|² − 2·a⃗·b⃗ follows by expanding with the binomial formula; for a⃗·b⃗ = 0 (right angle) the classical Pythagoras remains. This connection is used constantly for distances between points and for magnitudes in plane and sphere equations.