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# The plane P1:4x+7y+4z+81=0 is rotated through a right angle about its line of intersection with the plane P2:5x+3y+10z-25=0.If the plane in its new position is denoted by P3, and then,What is the distance of the plane from the origin?Please give a detailed stepwise approach.

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
7 years ago
P3 will be parallel to
$(4,7,4)*((4,7,4)*(5,3,10))= (-81,324,-486 )$
a point of intersection of two planes
$4x+7y=-4z-81\\ 5x+3y=-10z+25\\ x=\begin{pmatrix} -4z-81 &7 \\ -10z+25& 3 \end{pmatrix}/\begin{pmatrix} 4 &7 \\ 5 &3 \end{pmatrix} \\y=\begin{pmatrix} 4 &-4z-81 \\ 5& -10z+25 \end{pmatrix}/\begin{pmatrix} 4 &7 \\ 5 &3 \end{pmatrix}$
(x,y,z)
Now we just
$\vec a=(x,y,z)\\ distance\,from\,origin=\vec r.\vec n/|\vec n|=\vec a.\vec n/|\vec n|$
where n is perpendicular vector to plane we calculated a first
and a is (x,y,z)

Arun Kumar
IIT Delhi