Collinearity and Coplanarity of Vectors

Two vectors a and b are collinear if there exists a scalar λ such that a = λb.

Three points A(a), B(b) and C(c) are collinear iff ∃ non-zero scalars x, y, z such that xa + yb + zc = 0, where x + y + z = 0.

Three points A(a), B(b) and C(c) are collinear if AB × BC = 0 i.e. a × b + b × c + c × a = 0.

a and b are two non-collinear vectors. A vector r is coplanar with a and b if and only if there exists a unique scalar λ₁ and λ₂ such that r = λ₁a + λ₂b

Three vectors a₁i + a₂j + a₃k, b₁i + b₂j + b₃k and c₁i + c₂j + c₃k are coplanar, if \[\begin{vmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{vmatrix}=0.\]

Four points with position vectors a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k, c = c₁i + c₂j + c₃k and d = d₁i + d₂j + d₃k will be coplanar iff

\[\begin{vmatrix} a_1 & a_2 & a_3 & 1 \\ b_1 & b_2 & b_3 & 1 \\ c_1 & c_2 & c_3 & 1 \\ d_1 & d_2 & d_3 & 1 \end{vmatrix}=0.\]

In general, if a₁, a₂, …, aₙ are n vectors and t₁, t₂, …, tₙ are n scalars, then linear combination of vectors a₁, a₂, …, aₙ is t₁a₁ + t₂a₂ + … + tₙaₙ.

• For 2 vectors:

  \[\overline{\mathbf{r}}=x\overline{\mathbf{a}}+y\overline{\mathbf{b}}\]

• For 3 vectors:

  \[\mathbf{\overline{r}}=x\mathbf{\overline{a}}+y\mathbf{\overline{b}}+\mathbf{z}\mathbf{\overline{c}}\]