​ Let the vectors (2 + a + b)i (a + 2b + c)j (b - c)k (1 + b)i + 2bj - bk and (2 + b)i + 2bj (1 - b)k a, b, c, ∈ R be co-planar. ​

Question

Let the vectors (2 + a + b)\(\hat i\) (a + 2b + c)\(\hat j\) (b - c)\(\hat k\)  (1 + b)\(\hat i\) + 2b\(\hat j\) - b\(\hat k\)  and (2 + b)\(\hat i\) + 2b\(\hat j\) (1 - b)\(\hat k\)  a, b, c, ∈ R be co-planar. 

Then which of the following is true? 

(1) 2b = a + c 

(2) 3c = a + b 

(3) a = b + 2c 

(4) 2a = b + c

Answer 1

If the vectors are co-planar,

Now R₃ → R₃ – R₂, R₁ → R₁ – R₂

_{So \(\begin{bmatrix} a+1 & a + c & -c \\[0.3em] b+1 & 2b & -b \\[0.3em] 1 & 0 & 1 \end{bmatrix} \)} 

= (a + 1) 2b – (a + c) (2b + 1) – c(–2b) 

= 2ab + 2b – 2ab – a – 2bc – c + 2bc 

= 2b – a - c = 0