Projection onto a plane

Geometry Level 3

An orthogonal projection operator projects points of 3D space onto a plane that passes through the origin. If the point ( 5 , 3 , 5 ) (5,3,5) is projected into the point ( 3 , 5 , 1 ) (3,5,1) , what is the projection of ( 4 , 0 , 7 ) (4,0,7) ?

( 0 , 0 , 0 ) (0, 0, 0) ( 4 , 4 , − 2 ) (4, 4, -2) ( 5 , 1 , − 2 ) (5, 1, -2) ( 1 , 3 , 1 ) (1, 3, 1)

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Michael Mendrin
Jun 18, 2014

Let A = ( 5 , 3 , 5 ) , B = ( 3 , 5 , 1 ) A=(5,3,5), B=(3,5,1) so that C = A − B = ( − 2 , 2 , 4 ) C=A-B=(-2,2,4) . Then if D = ( 4 , 0 , 7 ) D=(4,0,7) and E E is the projection of D D on the plane, then for some scalar x x it has to be true that D + x C = E D+xC=E . Of the multiple choices offered, only D + ( 3 / 2 ) C = E D+(3/2)C=E is possible, where E = ( 1 , 3 , 1 ) E=(1,3,1) . This saves the trouble of directly computing E E , because of the limited choices being offered.

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