If a = i + j + k , b = i , c = c₁ i + c₂ j + c₃ k with c₁ = 1 , c₂ = 2 , then value of c₃ such that a , b , c are coplanar is ____

Step-by-step solution

Given vectors are a = i + j + k , b = i , and c = i + 2 j + c₃ k . For the vectors to be coplanar, their scalar triple product must be zero. This means the determinant of their components is zero: vmatrix 1 & 1 & 1 1 & 0 & 0 1 & 2 & c₃ vmatrix = 0 Expanding the determinant along the second row: -1(c₃ - 2) = 0 c₃ = 2 Answer: 2