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

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Answer 1 · 2017-01-09T14:19:35

The reqd. unit normal #i/sqrt2-j/sqrt2#.

The given points #P, Q, and, R# are co-planer; therefore so are the

vectors #vec(PQ) and vec(QR).#

Knowing that #vec n=vec(PQ) xx vec(QR)# is orthogonal to both of

these vectors, #vec n# is the normal vector of the plane containing

the points #P,Q, &, R#.

Now, #vec(PQ)=(5,-1,2)-(3,-3,0)=(2,2,2)=2(1,1,1),#

#vec(QR)=(0,0,4)=4(0,0,1)#

#vecn=8|(i,j,k),(1,1,1),(0,0,1)|=8(i-j) rArr ||vecn||=8sqrt2.#

Hence, the reqd. unit normal #hatn=vecn/||vecn||=i/sqrt2-j/sqrt2#.

Enjoy Maths.!