Class 12

Math

3D Geometry

Three Dimensional Geometry

Find the acute angle between the following planes.$2x−y+z=5$ and $x+y+2z=7$.

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A line passes through the points $(6,−7,−1)and(2,−3,1)˙$ Find te direction cosines off the line if the line makes an acute angle with the positive direction of the x-axis.

A mirror and a source of light are situated at the origin 0 and at a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the direction ratios of the normal to the plane are 1, -1, 1, then direction consines of the reflected rays are

Find the shortest distance between the lines $r=(1−λ)i^+(λ−2)j^ +(3−2λ)k^andr=(μ+1)i^+(2μ+1)k^˙$

Find the coordinates of a point on the $2x−1 =−3y+1 =z$ atg a distance $414 $ from the point $(1,−1,0)˙$

A variable plane passes through a fixed point $(a,b,c)$ and cuts the coordinate axes at points $A,B,andC˙$ Show that eh locus of the centre of the sphere $OABCisxa +yb +zc =2.$

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.

What is the angle between the planes2x−y+z=6 andx+y+2z=3?

Find the angle between the lines whose direction cosines are connected by the relations $l+m+n=0and2lm+2nl−mn=0.$