Question 138 of 379MediumMCQ
If a is a nonzero vector such that its projections on the vectors 2i^−j^+2k^, i^+2j^−2k^, and k^ are all equal, then a unit vector along a is: Solutions · 1
Official solution
Equal projections give:
\[\frac{\vec a\cdot(2\hat i-\hat j+2\hat k)}3 = \frac{\vec a\cdot\hat k}1 = \frac{\vec a\cdot(\hat i+2\hat j-2\hat k)}3\]
This yields a⋅(2i^−j^−k^)=0 and a⋅(i^+2j^−5k^)=0, so a is perpendicular to both, hence
\[\vec a \parallel (2\hat i-\hat j-\hat k)\times(\hat i+2\hat j-5\hat k) = (5+2,\,-1+10,\,4+1) = (7,9,5)\]
\[\hat a = \frac1{\sqrt{155}}(7\hat i+9\hat j+5\hat k)\] Answer the question to unlock