The distance of the point Q(0,2,−2) from the line passing through the point P(5,−4,3) and perpendicular to the lines r=(−3i^+2k^)+λ(2i^+3j^+5k^) and r=(i^−2j^+k^)+μ(−i^+3j^+2k^) is:
Solutions · 1
Official solution
Direction of the required line: (2,3,5)×(−1,3,2)=(−9,−9,9)=−9(1,1,−1), i.e. (1,1,−1).
Line through P(5,−4,3) with direction (1,1,−1). PQ=(−5,6,−5).
\[\vec{PQ}\times(1,1,-1)=(-1,-10,-11),\qquad |\vec{PQ}\times(1,1,-1)|^2=1+100+121=222.\]
\[\text{distance}^2=\frac{222}{3}=74\implies\text{distance}=\sqrt{74}.\]