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 ________

Question

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 ________,

MARKS App · Accepted Answer

a + 1 a a a a + 1 a a a a + 1 = 0 ⇒ a + 1 + a + a = 0 ⇒ a = - 1 3 P → = 2 3 i ^ - 1 3 j ^ - k ^ Q → = 1 3 - i ^ + 2 j ^ - k ^ R → = 1 3 - i ^ - j ^ + 2 k ^ P → . Q → = 1 9 - 2 - 2 + 1 = - 1 3 R → × Q → = 1 9 i j k - 1 2 - 1 - 1 - 1 2 = 1 9 i 4 - 1 - j - 2 - 1 + k 1 + 2 = 1 9 3 i + 3 j + 3 k = i + j + k 3 R → × Q → = 1 3 3 ⇒ R → × Q → 2 = 1 3 3 P → . Q → 2 - λ R → × Q → 2 = 0 3 . 1 9 - λ . 1 3 = 0 ⇒ λ = 1

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