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JEE Main 2025 · Mathematics · Vector Algebra · medium

Let vec a=hat i+hat j and vec b=hat j+hat k. The magnitude of the scalar projection of vec p=2hat i+3hat j+5hat k on a vector perpendicular to both…

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Let a⃗=i^+j^\vec a=\hat i+\hat j and b⃗=j^+k^\vec b=\hat j+\hat k. The magnitude of the scalar projection of p⃗=2i^+3j^+5k^\vec p=2\hat i+3\hat j+5\hat k on a vector perpendicular to both a⃗\vec a and b⃗\vec b is
  1. A. 434\sqrt3
  2. B. 23\dfrac{2}{\sqrt3}
  3. C. 43\dfrac{4}{\sqrt3}
  4. D. 43\dfrac43
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Answer: (C) 43\dfrac{4}{\sqrt3}

A vector perpendicular to both a⃗\vec a and b⃗\vec b is n⃗=a⃗×b⃗=∣i^j^k^110011∣=i^−j^+k^\vec n=\vec a\times\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\1&1&0\\0&1&1\end{vmatrix}=\hat i-\hat j+\hat k, with ∣n⃗∣=3|\vec n|=\sqrt3. The scalar projection of p⃗\vec p on n⃗\vec n is p⃗⋅n⃗∣n⃗∣=2−3+53=43\dfrac{\vec p\cdot\vec n}{|\vec n|}=\dfrac{2-3+5}{\sqrt3}=\dfrac4{\sqrt3}. The direction (sign) of n⃗\vec n does not matter for the magnitude. Using p⃗⋅n⃗=4\vec p\cdot\vec n=4 without dividing by ∣n⃗∣|\vec n| gives the vector component's coefficient, not the scalar projection.
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