Class 12

Math

3D Geometry

Three Dimensional Geometry

Find the length and the foot of the perpendicular drawn from the point $(1,1,2)$ to the plane $r⋅(2i^−2j^ +4k^)+5=0$.

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The point P is the intersection of the straight line joining the points $Q(2,3,5)$ and $R(1,−1,4)$ with the plane $5x−4y−z=1$. If S is the foot of the perpendicular drawn from the point $T(2,1,4)$ to QR, then the length of the line segment PS is (A) $2 1 $ (B) $2 $ (C) $2$ (D) $22 $

Shortest distance between the lines $1x−1 =1y−1 =1z−1 and1x−2 =1y−3 =1z−4 $ is equal to a. $14 $ b. $7 $ c. $2 $ d. none of these

Find the equation of the plane passing through $A(2,2,−1),B(3,4,$ $2)andC(7,0,6)˙$ Also find a unit vector perpendicular to this plane.

If the distance of the point $P(1,−2,1)$ from the plane $x+2y−2z=α,whereα>0,is5,$ then the foot of the perpendicular from $P$ to the place is a. $(38 ,34 ,−37 )$ b. $(34 ,−34 ,31 )$ c. $(31 ,32 ,310 )$ d. $(32 ,−31 ,−35 )$

Let $A(1,1,1),B(2,3,5)andC(−1,0,2)$ be three points, then equation of a plane parallel to the plane $ABC$ which is at distance $2$ is a. $2x−3y+z+214 =0$ b. $2x−3y+z−14 =0$ c. $2x−3y+z+2=0$ d. $2x−3y+z−2=0$

A parallelepiped is formed by planes drawn through the points $P(6,8,10)and(3,4,8)$ parallel to the coordinate planes. Find the length of edges and diagonal of the parallelepiped.

The Cartesian equation of a line is $2x−3 =−2y+1 =5z−3 $ . Find the vector equation of the line.

Find the plane of the intersection of $x_{2}+y_{2}+z_{2}+2x+2y+2=0and4x_{2}+4y_{2}+4z_{2}+4x+4y+4z−1=0.$