Consider two vectors in R³, u = (u₁, u₂, u₃)^T and v = (v₁, v₂, v₃)^T.

• The dot product of u with itself is u·u = u₁² + u₂² + u₃². For example, u = (3, 0, –1)^T, u·u = 3² + 0² + (–1)² = 9 + 0 + 1 = 10, and v = (2, 2, 1)^T, v·v = 2² + 2² + 1² = 4 + 4 + 1 = 9.
• This quantity is also the square of the norm (or length), u·u = ||u||², so the norm or length is ||u|| = √u·u. For example, the norm or length of u is ||u|| = √10, while the norm or length of v is ||v|| = √9=3.
• The dot product of u with v is u·v = u₁v₁ + u₂v₂ + u₃v₃. For example, u·v = 3(2) + 0(2) + –1(1) = 6 + 0 – 1 = 5.
• The cosine of the angle between u and v is given by cosθ = u·v / (||u|| ||v||). For example, cosθ = 5 / 3√10 = 5√10 / 30 = √10/6.
• The projection of v on u is given by Proj_u(v) = (u·v / u·u) u. For example, Proj_u(v) = (5/10) (3, 0, –1)^T = (3/2, 0, –1/2)^T. Similarly, Proj_v(u) = (u·v / v·v) v = (5/9) (2, 2, 1)^T = (10/9, 10/9, 1/9)^T.