The resultant of P⃗ and Q⃗ is perpendicular to P⃗. If Q is doubled without changing its direction, can the new resultant P⃗+2Q⃗ be perpendicular to the original Q⃗ for non-zero P and Q?
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Question

The resultant of P⃗ and Q⃗ is perpendicular to P⃗. If Q is doubled without changing its direction, can the new resultant P⃗+2Q⃗ be perpendicular to the original Q⃗ for non-zero P and Q?

G1 Physics · Accepted Answer

First condition: P+Q cosθ=0 ⇒ cosθ=−P/Q. Second condition would require 2Q+P cosθ=0. Substitution gives 2Q²−P²=0, which would imply P/Q=√2, but the first condition would then require cosθ=−√2, impossible. Hence no physical angle satisfies both.