Here, you will learn definition of collinear vectors, coplanar vectors, co-initial vectors and test of collinearity of three points.

Let’s begin –

## Definition of Collinear Vectors

Two vectors are said to be collinear if their supports are parallel disregards to their direction. Collinear vectors are also called **Parallel vectors**. If they have the same direction they are named as like vectors otherwise unlike vectors.

Symbolically, If \(\vec{a}\) & \(\vec{b}\) are collinear or parallel vectors, then there exists a scalar \(\lambda\) such that \(\vec{a}\) = \(\lambda\vec{b}\) or, \(\vec{b}\) = \(\lambda\vec{a}\).

**Theorem 1 :**

Two non-zero vectors \(\vec{a}\) & \(\vec{b}\) are collinear iff there exist scalars x, y not both zero such that x\(\vec{a}\) + y\(\vec{b}\) = \(\vec{0}\).

**Theorem 2 :**

If \(\vec{a}\) & \(\vec{b}\) are two non-zero non-collinear vectors and x, y are scalars then x\(\vec{a}\) + y\(\vec{b}\) = 0 \(\implies\) x = y = 0.

**Example** : If \(\vec{a}\) and \(\vec{b}\) are non-collinear vectors, find the value of x for which vectors \(\vec{\alpha}\) = (x – 2)\(\vec{a}\) + \(\vec{b}\) and \(\vec{\beta}\) = (3 + 2x)\(\vec{a}\) – 2\(\vec{b}\) are collinear.

**Solution** : Since vectors \(\vec{\alpha}\) and \(\vec{\beta}\) are collinear. Therefore, there exist scalar \(\lambda\) such that

\(\vec{\alpha}\) = \(\lambda\)\(\vec{\beta}\)

\(\implies\) (x – 2)\(\vec{a}\) + \(\vec{b}\) = \(\lambda\){(3 + 2x)\(\vec{a}\) – 2\(\vec{b}\)}

\(\implies\) {x – 2 – \(\lambda\)(3 + 2x)}\(\vec{a}\) + (1 + 2\(\lambda\)\(\vec{b}\) = \(\vec{0}\)

Now, given \(\vec{a}\) and \(\vec{b}\) are non-collinear.

Therefore, from theorem 2,

x – 2 – \(\lambda\)(3 + 2x) = 0 and 1 + 2\(\lambda\) = 0

x – 2 – \(\lambda\)(3 + 2x) = 0 and \(\lambda\) = \(-1\over 2\)

x – 2 + \(1\over 2\)(3 + 2x) = 0 \(\implies\) 4x + 1 = 0

\(\implies\) x = \(-1\over 4\)

## Test of Collinearity of three points in Vectors

(a) 3 points A B C will be collinear if \(\overrightarrow{AB}\) = \(\lambda\overrightarrow{BC}\), where \(\lambda\) \(\in\) R.

(b) Three points A, B, C with position vectors \(\vec{a}\),\(\vec{b}\),\(\vec{c}\) respectively are collinear, if & only if there exist scalars x,y,z not all zero simultaneously such that ; x\(\vec{a}\) + y\(\vec{b}\) + z\(\vec{c}\) = 0, where x + y + z = 0.

(c) Collinearity can also be checked by first finding the equation of line through two points and satisfy the third point.