The scalar product of the vector with a unit vector along the sum of vectors and is equal to one. Find the value of .
Let the two vectors be,
b → = 2 i ^ + 4 j ^ − 5 k ^ c → = λ i ^ + 2 j ^ + 3 k ^
The sum of vectors are,
b → + c → = 2 i ^ + 4 j ^ − 5 k ^ + λ i ^ + 2 j ^ + 3 k ^ b → + c → = ( 2 + λ ) i ^ + ( 4 + 2 ) j ^ + ( − 5 + 3 ) k ^ b → + c → = ( 2 + λ ) i ^ + 6 j ^ − 2 k ^
It is given that vector a → = i ^ + j ^ + k ^ has scalar product with a unit vector along b → + c → , then,
( i ^ + j ^ + k ^ ) ( 1 λ 2 + 4 λ + 44 × ( ( 2 + λ ) i ^ + 6 j ^ − 2 k ^ ) ) = 1 1 λ 2 + 4 λ + 44 ( 1 i ^ + 1 j ^ + 1 k ^ ) . ( ( 2 + λ ) i ^ + 6 j ^ − 2 k ^ ) = 1 1. ( λ + 2 ) + 1.6 + 1. ( − 2 ) = λ 2 + 4 λ + 44 λ + 6 = λ 2 + 4 λ + 44
Squaring both sides of the above equation,
( λ + 6 ) 2 = λ 2 + 4 λ + 44 λ 2 + 12 λ + 36 = λ 2 + 4 λ + 44 λ 2 − λ 2 + 12 λ − 4 λ = 44 − 36 λ = 1
Thus, the value of λ = 1 .
Let the two vectors be,
The sum of vectors are,
It is given that vector
Squaring both sides of the above equation,
Thus, the value of
