Scalar multiples and components

λa is a vector of magnitude |λ||a|, along a when λ > 0 and opposite to a when λ < 0; 0·a = 0 and k·0 = 0. With λ = 1/|a| (a ≠ 0), λa is the unit vector â = a/|a|.

î, ĵ, k̂ are unit vectors along the x-, y- and z-axes. Every vector can be written in component form r = xî + yĵ + zk̂, with |r| = √(x² + y² + z²); x, y, z are its scalar components and xî, yĵ, zk̂ its vector components.

Two vectors are equal exactly when their î, ĵ and k̂ components are equal. Sum, difference and scalar multiples work component by component: (a₁ + b₁)î + (a₂ + b₂)ĵ + (a₃ + b₃)k̂ and so on.

Collinear test: b and a are collinear when b = λa, that is when b₁/a₁ = b₂/a₂ = b₃/a₃ = λ (with the usual care over zero components).

For a = a₁î + a₂ĵ + a₃k̂ the direction ratios are a₁, a₂, a₃ and the direction cosines are a₁/|a|, a₂/|a|, a₃/|a|.

Worked: the unit vector along 2î + 3ĵ + k̂ is (2î + 3ĵ + k̂)/√14. A vector of length 7 along î − 2ĵ is 7(î − 2ĵ)/√5 = (7/√5)î − (14/√5)ĵ.

Worked: the sum of 2î + 2ĵ − 5k̂ and 2î + ĵ + 3k̂ is 4î + 3ĵ − 2k̂, of length √29, so the unit vector along it is (4î + 3ĵ − 2k̂)/√29. For î + ĵ − 2k̂ the direction cosines are 1/√6, 1/√6, −2/√6.