Question

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are `55 hat i - 2 hat j, 5 hat i + 8 hat j and a hat i - 52 hat j` respectively, find the value of ‘a’.

Answer 1

`vec A = (55hat i - 2 hat j)`

`vec B = (5hat i + 8 hat j)`

`vec C = (a hat i − 52hat j)`

If three points are collinear, the vectors formed between them must be parallel. Let‘s find vectors `vec ("AB") and vec("BC")`

`vec ("AB") = (5 - 55) hat i + (8 - (-2)) hat j`

= `-50 hat i + 10 hat j`

`vec ("BC") = (a - 5) hat i + (-52 - 8) hat j`

= `(a - 5) hat i - 60 hat j`

For the vectors to be parallel the ratio of their corresponding components must be equal:

`(a - 5)/-50 = (-60)/10`

`(a - 5)/-50 = -6`

a − 5 = −6 × (−50)

a − 5 = 300

a = 300 + 5

a =305