Class 12

Math

3D Geometry

Three Dimensional Geometry

Shortest distance between the lines:$r=(4i^−j^ )+λ(i^+2j^ −3k^)$ and $r=(i^−j^ +2k^)+u(2i^+4j^ −5k^)$

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Show that the lines $−3x+1 =2y−3 =1z+2 $ and $1x =−3y−7 =2z+7 $ are coplanar. Find the equation of the plane containing these lines.

Show that the equation $ax+by+d=0$ represents a plane parallel to the z-axis. Hence, find the equation of a plane which is parallel to the z-axis and passes through the points $A(2,−3,1)$ and $B(−4,7,6)$.

Find the equation of the plane which contains two parallel lines given by $1x−3 =−4y+2 =5z $ and $1x−4 =−4y−3 =5z−2 $.

Find the distance of the point $(1,1,2)$ from the plane $r⋅(2i^−2j^ +4k^)+5=0$.

The direction ratios of two lines are $3,2,−6$ and $1,2,2$ respectively. The acute angle between these lines are

Find the equation of a plane passing through the points $A(a,0,0),B(0,b,0)$ and $C(0,0,c)$.

What is the angle between the vector $r=(4i^+8j^ +k^ )$ and the x-axis?

Find the equation of the plane passing through the intersection of the planes $4x−y+z=10$ and $x+y−z=4$ and parallel to the line with direction ratios $2,1,1$. Find also the perpendicular distance of $(1,1,1)$ from this plane.