Question

If the points A( 5, 5, $\lambda $) , B ( - 1, 3 , 2) and C ( -4 , 2 , -2) are collinear, find the value of $\lambda $.

Answer 1

Hint – In order to solve this problem use the concept of direction ratios and show then proportional doing this will solve your problem.

Complete step-by-step answer:
As we know the three points are collinear it means the points are on the same line.
We also know that if the points are collinear then their directions ratios are proportional.
Therefore the three given points are A( 5, 5, $\lambda $) , B ( - 1, 3 , 2) and C ( -4 , 2 , -2).
The directions ratio (DR) of AB is $( - 1 - 5)\hat i + (3 - 5)\hat j + (2 - \lambda )\hat k = - 6\hat i - 2\hat j + (2 - \lambda )\hat k...........(1)$
The DRs of BC is $( - 4 - ( - 1))\hat i + (2 - 3)\hat j + ( - 2 - 2)\hat k = - 3\hat i - \hat j - 4\hat k...........(2)$
As we know that (1) and (2) are proportional.
So, we do: $\dfrac{{ - 6}}{{ - 3}} = \dfrac{{ - 2}}{{ - 1}} = \dfrac{{2 - \lambda }}{{ - 4}} = 2$
So, we can say that,
$
  2 - \lambda = - 8 \\
  \lambda = 10 \\
$
Hence, the value of $\lambda = 10$.

Note – To solve this problem we have found the direction ratios and obtained those vectors then we have used the concept that if the points are collinear then their direction ratios are proportional. With the help of that we have calculated the value of $\lambda $ and got the answer.