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.
