1If x? and ? be two non-zero vectors such that |x? + ?| = |x?| and 2x? + λ? is perpendicular to ?, then the value of λ is
1If x? and ? be two non-zero vectors such that |x? + ?| = |x?| and 2x? + λ? is perpendicular to ?, then the value of λ is
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1 Answer
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|x + y|² = |x|²
(x+y)· (x+y) = x·x
|x|² + 2x·y + |y|² = |x|²
|y|² + 2x·y = 0 (1)
and (2x + λy)·y = 0
2x·y + λ|y|² = 0 (2)
From (1), 2x·y = -|y|².
Substitute into (2):
-|y|² + λ|y|² = 0
(λ-1)|y|² = 0
Assuming y is a non-zero vector, |y|² ≠ 0, therefore λ=1.
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