Question

Q1.The vectors λi + j + 2k, i + λj – k and 2i – j + λk are coplanar if

Answer 1

Step-by-step explanation:

We have given that, λi + j + 2k, i + λj – k and 2i – j + λk are coplanar.

we have to find the value of λ

solution : if three vectors a, b and c are coplanar, then [a b c ] = 0

$\begin{gathered}\left|\begin{array}{ccc}\lambda &1&2\\1&\lambda&-1\\2&-1&\lambda\end{array}\right|=0\end{gathered} </p><p>$

⇒ λ(λ² - 1) - 1(λ + 2) + 2(-1 - 2λ) = 0

⇒λ³ - λ - λ - 2 -2 - 4λ = 0

⇒λ³ - 6λ - 4 = 0

⇒λ³ + 2λ² - 2λ² - 4λ - 2λ - 4 = 0

⇒λ²(λ + 2) - 2λ(λ + 2) - 2(λ + 2) = 0

⇒(λ + 2)(λ² - 2λ + 2) = 0

here λ = -2 , and (λ² - 2λ + 2) = 0 doesn't have real solution.

so for the value of λ = -2, all three vectors are coplanar.