Let us understand that, two more points are said to be collinear if they all lie on a single straight line.
Let the points be A, B and C having position vectors such that,
Position vector of A = \(10\hat i+3j\)
Position vector of B = \(12\hat i-5j\)
Position vector of C = \(a\hat i+11\hat j\)
So, let us find \(\vec {AB}\) and \(\vec {BC}.\)
Therefore, \(\vec {AB}\) is given by
\(\vec {AB}\) = Position vector of B - Position vector of A
Since, it has been given that points A, B and C are collinear.
So, we can write as
\(\vec {BC}=\lambda\vec{AB}\)
Where λ = a scalar quantity
Put the values of \(\vec {BC}\) and \(\vec {AB}\) from (i) and (ii), we get
Comparing the vectors \(\vec i\) and \(\vec j\) respectively, we get
a – 12 = 2λ …(iii)
and, 16 = –8λ
From –8λ = 16, we can find the value of λ.
–8λ = 16
⇒ \(\lambda=-\cfrac{16}8\)
⇒ λ = –2
Put λ = –2 in equation (iii), we get
a – 12 = 2λ
⇒ a – 12 = 2(–2)
⇒ a – 12 = –4
⇒ a = –4 + 12
⇒ a = 8
Thus, we have got a = 8.
