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NDA 2023 Maths · Mathematics · Vector Algebra · medium

If vec a and vec b are non-zero vectors with |vec a+vec b|=|vec a-vec b|, then the angle between vec a and vec b is

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If a⃗\vec a and b⃗\vec b are non-zero vectors with ∣a⃗+b⃗∣=∣a⃗−b⃗∣|\vec a+\vec b|=|\vec a-\vec b|, then the angle between a⃗\vec a and b⃗\vec b is
  1. A. 0∘0^\circ
  2. B. 60∘60^\circ
  3. C. 180∘180^\circ
  4. D. 90∘90^\circ
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Answer: (D) 90∘90^\circ

Squaring, ∣a⃗∣2+∣b⃗∣2+2a⃗⋅b⃗=∣a⃗∣2+∣b⃗∣2−2a⃗⋅b⃗|\vec a|^2+|\vec b|^2+2\vec a\cdot\vec b=|\vec a|^2+|\vec b|^2-2\vec a\cdot\vec b, so a⃗⋅b⃗=0\vec a\cdot\vec b=0 and the vectors are perpendicular. In a parallelogram this says that the diagonals are equal, i.e. it is a rectangle.
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