Consider the vectors x = i ^ + 2 j ^ + 3 k ^ , y = 2 i ^ + 3 j ^ + k ^ , and z = 3 i ^ + j ^ + 2 k ^ For two distinct positive real numbers α and β , define X = α x + β y − z , Y = α y + β

Question

Consider the vectors x = i ^ + 2 j ^ ​ + 3 k ^ , y ​ = 2 i ^ + 3 j ^ ​ + k ^ , and z = 3 i ^ + j ^ ​ + 2 k ^ For two distinct positive real numbers α and β , define X = α x + β y ​ − z , Y = α y ​ + β z − x , and Z = α z + β x − y ​ If the vectors X , Y , and Z lie in a plane, the value of α + β − 3 is ______ .

TestHub · Accepted Answer

⇒ ​ α − 1 β ​ β α − 1 ​ − 1 β α ​ ​ − 0 ​ 1 2 3 ​ 2 3 1 ​ 3 1 2 ​ ​ ​ ​ = 0 ​ ⇒ ( α 3 + β 3 − 1 ) − ( − α β − α β − α β ) = 0 ⇒ α 3 + β 3 + 3 α β = 1 ⇒ α 3 + β 3 + ( − 1 ) 3 = 3 ( α ) ( β ) ( − 1 ) ⇒ α + β − 1 = 0 ​ So, α + β − 3 = − 2

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