Conditions for Collinearity of Vectors

We have learnt that two or more vectors that point in same or opposite directions and parallel to each other are called collinear vectors.

In mathematics, we have certain conditions that must be satisfied by two or more vectors to be considered collinear. Consider two vectors [Tex]\overrightarrow{A}[/Tex] and [Tex]\overrightarrow{B}[/Tex] .The conditions for collinearity of vectors are as follows:

Condition 1: If [Tex]\overrightarrow{A} = n \overrightarrow{B}[/Tex], where n is any scalar then vectors A and B are said to be collinear.

Condition 2: If the ratio of the corresponding coordinates of two vectors are equal, then they are said to be collinear. This condition is not applicable if any one of the coordinates of any vector is zero. Consider [Tex]\overrightarrow{A} = a\hat{i}+b\hat{j}+c\hat{k} [/Tex]and [Tex]\overrightarrow{B} = p\hat{i}+q\hat{j}+r\hat{k} [/Tex], then they are said to be collinear if:

[Tex]\bold{\frac{a}{p}=\frac{b}{q}=\frac{c}{r}}[/Tex] OR [Tex]\bold{\frac{p}{a}=\frac{q}{b}=\frac{r}{c}}[/Tex]

Condition 3: Two vectors are said to be collinear if their cross product is zero i.e. [Tex]\overrightarrow{A} \times \overrightarrow{B} = 0[/Tex].

Read More about Collinear Points.

Vectors are also called Euclidean vectors or Spatial vectors, and they have many applications in mathematics, physics, engineering, and various other fields. There are different types of vectors, including zero vectors (which have 0 magnitude and no direction), unit vectors (which have a magnitude of 1), position vectors, co-initial vectors, like and unlike vectors, co-planar vectors, collinear vectors, equal vectors, displacement vectors, and negative vectors.

In this article, we will discuss collinear vectors and the criteria according to which two vectors are said to be collinear in detail.