Question

Find the values of ‘a’ for which the vectors α = i + 2j + k, β = ai + j + 2k and γ = i + 2j + ak are coplanar.

Answer 1

Three vectors are coplanar if (if and only if)  \(\vec a.(\vec b\times\vec c)=0\)

Hence we have value of the matrix \(\begin{vmatrix} 1 & 2 & 1 \\ a & 1 & 2 \\ 1 & 2 & a \end{vmatrix}\) = 0

We have 2a² - 3a + 1 = 0

2a² - 2a - a + 1 = 0

Solving this quadratic equation we get

a = 1, a = \(\cfrac12\)