The term collinearity means points lie on the same line whether they are close together, far apart or form a ray, line segment or line. In other words, the set of points is said to be collinear if there exists a linear relation between them, such that the sum of the coefficient in it is zero. 

For example, collinear points can be shown in geometric figures such as ray, line, line segment, triangles, quadrilaterals, parallelograms and many more.

Collinear points

In the above figure the points A, B and C lie on the same line, so they are said to be collinear.

Note: We cannot draw a circle with three collinear points. We can draw only one circle from three non-collinear points.

There are numerous methods to find collinearity of three points, including slope method, distance formula and area of triangular formula. Now we shall learn about the necessary and sufficient condition of collinearity of three points using one of the above methods.

Essential Conditions of collinearity of three points:

The three points are said to be collinear if their slopes are equal.

Let P(x1,y1,z1), Q (x2,y2,z2) and R(x3,y3,z3) be any three points.

Plane 

The equation of line passes through points Px1,y1,z1and Q (x2,y2,z2) are

x-x1x2–x1=y-y1y2–y1=z-zz2–z1

Points P, Q, R will be collinear if R lies on the line through P and Q,

i.e. if R(x3,y3,z3) satisfies equation(1)

i.e.  if x3–x1x2–x1=y3–y1y2–y1=z3–z1z2–z1

This is the required condition for three points x1,y1,z1, x2,y2,z2, (x3,y3,z3) to be collinear.

Collinearity of three points in vector form: The set of points are said to be collinear in vector form if there exists a linear relation between them, such that the sum of the coefficients in it is zero. 

Necessary and sufficient condition of collinearity of three points in vector form:
let a, b , c be the position vectors of three given points.

The equation line passing through the points b and  c is

r= b+ t(c– b)

The points a, b , c will be collinear

If a= b+ t(c– b)

i.e. if a– b– tc– b=0

a+(t-1) b-tc=0

the algebraic sum of coefficients =1+t-1+-t=1+t-1-t=0

This is known as the collinearity of three points in vector form.

We have discussed the above Collinear points and the condition of collinearity in scalar form. Now we will discuss collinear vectors. 

Collinear vectors

The vectors are said to be collinear when they lie on the same given line. Two or more vectors are said to be collinear only if they are parallel to each other in the same or opposite direction.

                                       Collinear vectors in                            Collinear vectors

                                       The same direction                       in the same direction

Conditions of Collinear Vectors

• Two vectors a and  b are said  to be collinear vectors if there exists a scalar constant ‘k’ such that  a=kb   

• Two vectors a and  b are said to be collinear vectors if the ratio of their coordinates is equal. This condition does not exist in the case that one of the components of the given vector is 0.

• Two vectors a and  b are said to be collinear vectors if their cross product is equal to the zero vector. This condition can be applied only to three-dimensional problems.

Solved examples

Q.1 Prove that the points (1,2,3), (4,0,4), (-2,4,2), (7, -2,5) are collinear.

Solution:  Let A(1,2,3), B(4,0,4), C(-2,4,2), D(7, -2,5) be the given points.

The equations of AB are:

x-14-1=y-20-2=z-34-3

x-13=y-2-2=z-31

C  lies on  (1) if

-2-13=4-2-2=2-31

If

  -1=-1=-1, which is true.

∴C lies on AB

Similarly, D lies on AB

Hence A, B, C, D are Collinear.

Q2. Show that the points whose position vectors are given by-2i+3j +5k, i+2j +3k, 7i–k are collinear.

Solution: given points have position vectors as -2i+3j +5k, i+2j +3k and 7i–k 

∴ points are (-2, 3,5), (1,2,3), (7,0,-1)

The equation of straight line through (-2, 3,5), (1,2,3) is 

x+21+2=y-32-3=z-53-5

x+23=y-3-1=z-5-2

The point (7,0,-1) will lie on it

If 7+23=0-3-1=-1-5-2

i.e. if

3=3=3, which is true.

∴The points (-2, 3,5), (1,2,3), (7,0,-1) are collinear.

Hence, the points whose position vectors are given by -2i+3j +5k, i+2j +3k, 7i–k are Collinear.

Q3. Show the given vectors are collinear.  A= (3,4,5),   B = (6,8,10).

Solution: Two vectors are said to be collinear vectors if the ratio of their coordinates is equal. 

A1/B1 = 3/6 = 1/2

A2/B2= 4/8 = 1/2

A3/B3 = 5/10 = 1/2

Since A1/B1 = A2/B2 =A3/B3, the vectors.   A  and B can be considered as collinear vectors.

Q4 Find if the given vectors are collinear. A = 3i + 2j + 4k, B = – 3i -2j –4k

Solution: Two vectors are said to be collinear if one vector is a scalar multiple of the other vector.

(B)= -3 i -2 j – 4k = – (3i + 2j + 4k) = -(A)

⇒ (B) is a scalar multiple of (A).

Hence proved.

Conclusion 

In this article, we have learned about collinearity, collinear points and their necessary and sufficient conditions. The term collinearity means points lie on the same line whether they are close together, far apart or form a ray, line segment or line. In other words, the set of points is said to be collinear if there exists a linear relation between them, such that the sum of the coefficient in it is zero.