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Question

Find λ and μ if .

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Solution

The given vectors are ( 2 i ^ +6 j ^ +27 k ^ )×( i ^ +λ j ^ +μ k ^ )=0 .

The cross product of two vectors ( a 1 i ^ + a 2 j ^ + a 3 k ^ ) and ( b 1 i ^ + b 2 j ^ + b 3 k ^ )is given by,

a × b= | i ^ j ^ k ^ a 1 a 2 a 3 b 1 b 2 b 3 | ( 2 i ^ +6 j ^ +27 k ^ )×( i ^ +λ j ^ +μ k ^ )=| i ^ j ^ k ^ 2 6 27 1 λ μ |=0 i ^ +0 j ^ +0 k ^

Simplify the determinant in equation (2),

i ^ ( 6μ27λ ) j ^ ( 2μ27 )+ k ^ ( 2λ6 )=0 i ^ +0 j ^ +0 k ^

Compare corresponding components,

6μ27λ=0 2μ27=0 2λ6=0

Now,

2λ6=0 λ= 6 2 λ=3

2μ27=0 μ= 27 2

Thus, if ( 2 i ^ +6 j ^ +27 k ^ )×( i ^ +λ j ^ +μ k ^ )=0 , then λ=3 and μ= 27 2 .


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