Question

If the vectors `ahat("i")+hat("j")+hat("k"),  hat("i")+bhat("j")+hat("k")` and `hat("i")+hat("j")+chat("k")` are coplanar (a ≠ b ≠ c ≠ 1), then the value of abc - (a + b + c) = ______.

Options

• 2

• -2

• 0

• -1

Answer 1

If the vectors `ahat("i")+hat("j")+hat("k"),  hat("i")+bhat("j")+hat("k")` and `hat("i")+hat("j")+chat("k")` are coplanar (a ≠ b ≠ c ≠ 1), then the value of abc - (a + b + c) = -2.

Explanation:

Since `ahat("i")+hat("j")+hat("k"),  hat("i")+bhat("j")+hat("k")` and `hat("i")+hat("j")+chat("k")` are coplanar,

`abs[[a,1,1],[1,b,1],[1,1,c]]=0`

⇒ a (bc - 1) - 1 (c - 1) + 1 (1 - b) = 0

⇒ abc - a - b - c + 2 = 0

⇒ abc - (a + b + c) = -2