Question

if the vectors A= ai+j+k , B=i+bj+k C=i+j+ck are coplanar . then show that (1/1-a) + (1/1-b)+(1/1-c) =1

Answer 1

Answer with Explanation :

Given

A = ai + j + k

B = i + bj + k

C = i + j + ck

[A + B + C ] = 0

a     1      1

1      b     1    =  0

1      1      c

Applying C₁ - C₂ and C₂ - C₃

a-1     0        1

1        b-1      1    =  0

1-c     1-c      c

By taking (a-1)(b-1)(1-c) common

here a ≠ b ≠ c ≠ 1

So, (a-1)(b-1)(1-c) ≠ 0

and

1      0      1/(a-1)

0      1       1/(b-1)   =   0

1       1       c/(1-c)

$(\frac{c}{1-c} - \frac{1}{b-1}) + \frac{1}{a-1}(-1) = 0$

Now simplify

$\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} = 1$

Hence proved

Answer 2

Hence this question is proved.