Projection and two inequalities

The projection of a on b is the signed length of a's shadow on the line of b: a·b̂ = (a·b)/|b| = |a| cos θ. The projection vector is (a·b̂)b̂. The projection is positive for acute θ, zero for θ = π/2 and negative for obtuse θ.

The direction cosines of a are its projections on the axes divided by |a|: cos α = a₁/|a| and so on. The projection of a on itself is |a|.

Worked: the projection of 2î + 3ĵ + 2k̂ on î + 2ĵ + k̂ is (2 + 6 + 2)/√6 = 10/√6 = 5√6/3.

Magnitudes of sums come from |a ± b|² = |a|² ± 2a·b + |b|². With |a| = 2, |b| = 3 and a·b = 4, |a − b| = √(4 + 9 − 8) = √5.

If â is a unit vector and (x − â)·(x + â) = 8, then |x|² − 1 = 8 and |x| = 3.

Cauchy–Schwarz inequality: |a·b| ≤ |a||b|, since |cos θ| ≤ 1. Triangle inequality: |a + b| ≤ |a| + |b|, with equality only when a and b point the same way.

Collinear points by lengths: A(−2, 3, 5), B(1, 2, 3), C(7, 0, −1) give |AB| = √14, |BC| = 2√14, |AC| = 3√14. Since |AB| + |BC| = |AC|, B lies on segment AC. The three points cannot be the corners of a triangle: the triangle inequality holds with equality.