Angle Between Vectors

Question

If two nonzero vectors a and b satisfy |a + b| = |a - b|, what can be concluded about the angle between vectors a and b?

RankGuru · Accepted Answer

The key tool is the relationship between the magnitude of a vector sum and the dot product: |u|^2 = u . u, so squaring both sides converts the geometric condition into an algebraic one. This identity is a standard JEE Advanced device for extracting angle information. Squaring |a + b| = |a - b| gives |a|^2 + 2 a . b + |b|^2 = |a|^2 - 2 a . b + |b|^2. Cancelling the common terms leaves 4 a . b = 0, hence a . b = 0. Since a and b are nonzero, |a||b| is positive, so cos(theta) = 0, which forces theta = 90 degrees, meaning the vectors are perpendicular. The parallel option would require a . b = |a||b| (positive maximum), contradicting a . b = 0. The anti-parallel option needs a . b = -|a||b|, again nonzero. The 60-degree option would give a . b = |a||b|/2, also nonzero, so all three are excluded. Geometrically, |a + b| = |a - b| means the diagonals of the parallelogram built on a and b are equal, which characterizes a rectangle, whose adjacent sides are perpendicular. Plausibility check: for a = i and b = j the condition reads sqrt(2) = sqrt(2), confirming perpendicular vectors satisfy it exactly.

Angle Between Vectors

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