The value of 'a' such that the vector 2î - ĵ + k̂, î + 2ĵ - 3k̂ and 3î + aĵ + 5k̂ are coplanar, is

Concept used:

a, b, and c vectors are coplanar when \([\vec{a}, \vec{b},\vec{c}] = 0\)

Calculation:

\(\left| {\begin{array}{*{20}{c}} {2}&{-1}&{1}\\ {1}&{2}&{-3}\\ {3}&{a}&{5} \end{array}} \right| = 0\)

⇒ 2(10 + 3a) + 1(5 + 9) + 1(a - 6) = 0

⇒ 20 + 6a + 14 + a - 6 = 0

⇒ 7a = - 28

∴ a = - 4

∴ The vector 2î - ĵ + k̂, î + 2ĵ - 3k̂ and 3î + aĵ + 5k̂ are coplanar when a = - 4