How do you find a unit vector with positive first coordinate that is orthogonal to the plane through the points #P = (3, -3, 0), Q = (5, -1, 2)#, and #R = (5, -1, 6)#?

1 Answer
Cesareo R.
Sep 8, 2016

#sqrt(2)/2{1,-1,0} #

Explanation:

If #P = (3, -3, 0), Q = (5, -1, 2)#, and #R = (5, -1, 6)# pertain to the plane #Pi#

then #(P-Q) xx (R-Q)# is normal to #Pi#

so

#vec n = (P-Q) xx (R-Q) = {-8, 8, 0}#. Any vector proportional to #vec n# like #lambda vec n# with #lambda in RR, lambda ne 0# is also normal to #Pi# so we choose

#lambda = -1/norm(vec n) = -1/(8 sqrt(2))#. The sought unit vector is then

#lambda vec n =sqrt(2)/2{1,-1,0} #

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