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NDA 2025 Maths · Mathematics · Vector Algebra · medium

If vec a,vec b,vec c are three mutually perpendicular unit vectors, then |vec a+vec b+vec c| is equal to

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If a⃗,b⃗,c⃗\vec a,\vec b,\vec c are three mutually perpendicular unit vectors, then ∣a⃗+b⃗+c⃗∣|\vec a+\vec b+\vec c| is equal to
  1. A. 3\sqrt3
  2. B. 33
  3. C. 11
  4. D. 2\sqrt2
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Answer: (A) 3\sqrt3

∣a⃗+b⃗+c⃗∣2=∣a⃗∣2+∣b⃗∣2+∣c⃗∣2+2(a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗)=1+1+1+0=3|\vec a+\vec b+\vec c|^2=|\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)=1+1+1+0=3, as the dot products of perpendicular vectors vanish. Hence the magnitude is 3\sqrt3. The value 33 is the square of the magnitude, and 2\sqrt2 would be for only two vectors.
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