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 &#8594; = a + 1 i ^ + a j ^ + a k ^ , q &#8594; = a i ^ + a + 1 j ^ + a k ^ and r &#8594; = a i ^ + a j ^ + a + 1 k ^ a &#8712; R are coplanar and 3 p &#8594; . q &#8594; 2 - &#955; r &#8594; &#215; q &#8594; 2 = 0 , then the value of &#955; is ________

Aicharya · Accepted Answer

a + 1 a a a a + 1 a a a a + 1 = 0 &#8658; a + 1 + a + a = 0 &#8658; 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 &#8658; R → × Q → 2 = 1 3 3 P → . Q → 2 - λ R → × Q → 2 = 0 3 . 1 9 - λ . 1 3 = 0 &#8658; λ = 1

If the vectors, p → = a + 1 i ^ + a j ^ + a k ^ , q → = a i ^ + a + 1 j ^ + a k ^ and r →…

Solution

If the vectors, p &#8594; = a + 1 i ^ + a j ^ + a k ^ , q &#8594; = a i ^ + a + 1 j ^ + a k ^ and r &#8594;…

Solution

If the vectors, p → = a + 1 i ^ + a j ^ + a k ^ , q → = a i ^ + a + 1 j ^ + a k ^ and r →…