JEE-Advanced 2015 (28/40)

Geometry Level 4

Suppose that p ⃗ , q ⃗ \vec p, \vec q and r ⃗ \vec r are three non-coplanar vectors in R 3 \mathbb R^3 . Let the components of a vector s ⃗ \vec s along p ⃗ , q ⃗ \vec p, \vec q and r ⃗ \vec r be 4, 3 and 5 respectively. If the components of this vector s ⃗ \vec s along ( − p ⃗ + q ⃗ + r ⃗ ) , ( p ⃗ − q ⃗ + r ⃗ ) (-\vec p+\vec q+\vec r),(\vec p- \vec q +\vec r) and ( − p ⃗ − q ⃗ + r ⃗ ) (-\vec p -\vec q +\vec r) are x , y x,y and z z respectively, then find the value of 2 x + y + z 2x+y+z .


The answer is 9.

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Harish Nandan
Oct 14, 2015

s⃗ =4p⃗ +3q⃗ +5r⃗ 

       s⃗ =x(−p⃗ +q⃗ +r⃗ )+y(p⃗ −q⃗ +r⃗ )+z(−p⃗ −q⃗ +r⃗ )

       s⃗ =(–x+y–z)p⃗ +(x–y–z)q⃗ +(x+y+z)r⃗ 

       ⇒ –x+y–z=4⇒ x–y–z=3⇒ x+y+z=5

       On solving we get x = 4, y=9/2, z=−7/2

       ⇒2x+y+z=9
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