If the vectors, p = (a + 1) i + a j + a k, q = a i + (a + 1) j + a k and r = a i + a j + (a + 1) k(a R) are coplanar and 3(p. q)² - λ | r × q |² = 0, then the value of λ is _ _ _.

Vector Algebra: JEE Main 2020 Mathematics question

JEE Main 2020 · 9 Jan, Morning shift · Q42

JEE Main 20209 Jan, Morning shiftMathematicsHardVector Algebra184× since 2002

If the vectors,

\overrightarrow p = \left( {a + 1} \right)\widehat i + a\widehat j + a\widehat k

\overrightarrow q = a\widehat i + \left( {a + 1} \right)\widehat j + a\widehat k

and

\overrightarrow r = a\widehat i + a\widehat j + \left( {a + 1} \right)\widehat k\left( {a \in R} \right)

are coplanar and

3{\left( {\overrightarrow p .\overrightarrow q } \right)^2} - \lambda \left| {\overrightarrow r \times \overrightarrow q } \right|^2 = 0

, then the value of

\lambda

is ______.

Your answerType your answer. Integers and decimals are accepted.