Find the value of \( \lambda \) so that the vector \( \vec{A} = 2\hat{i} + \lambda\hat{j} - \hat{k} \) and \( \vec{B} = 4\hat{i} - 2\hat{j} - 2\hat{k} \) are orthogonal to each other.
Answer (2)
To make the vectors A and B orthogonal, set their dot product equal to zero. Solving the equation yields λ = 5 .
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To find the value of λ such that the two vectors A = 2 i ^ + λ j ^ − k ^ and B = 4 i ^ − 2 j ^ − 2 k ^ are orthogonal, we need to use the concept of the dot product of vectors.
Two vectors are orthogonal if their dot product is zero.
The dot product of two vectors A = a 1 i ^ + a 2 j ^ + a 3 k ^ and B = b 1 i ^ + b 2 j ^ + b 3 k ^ is given by:
A ⋅ B = a 1 b 1 + a 2 b 2 + a 3 b 3
Applying this to our vectors A = 2 i ^ + λ j ^ − k ^ and B = 4 i ^ − 2 j ^ − 2 k ^ :
a 1 = 2 , a 2 = λ , a 3 = − 1
b 1 = 4 , b 2 = − 2 , b 3 = − 2
Substitute these values into the dot product formula:
A ⋅ B = ( 2 ) ( 4 ) + ( λ ) ( − 2 ) + ( − 1 ) ( − 2 )
Simplifying the expression, we get:
8 − 2 λ + 2 = 0
Combine the terms:
10 − 2 λ = 0
To find λ , solve the equation:
2 λ = 10 λ = 2 10 λ = 5
So, the value of λ that makes the vectors A and B orthogonal is λ = 5 .
