Class 12

Math

3D Geometry

Three Dimensional Geometry

If $r=(i^+2j^ +3k^)+λ(i^−j^ +k^)$ and $r=(i^+2j^ +3k^)+μ(i^+j^ −k^)$ are two lines, then the equation of acute angle bisector of two lines is

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The Cartesian equations of a line are $6x−2=3y+1=2z−2.$ Find its direction ratios and also find a vector equation of the line.

Find the values $p$ so that line $31−x =2p7y−14 =2z−3 and3p7−7x =1y−5 =56−z $ are at right angles.

Find the equation of the image of the plane $x−2y+2z−3=0$ in plane $x+y+z−1=0.$

If the line $x=y=z$ intersect the line $s∈Ax˙+s∈By˙ +s∈Cz˙=2d_{2},s∈2Ax˙+s∈2By˙ +s∈2Cz˙=d_{2},$ then find the value of $2sinA 2sinB˙ 2sinC˙ whereA,B,C$ are the angles of a triangle.

Find the point where line which passes through point $(1,2,3)$ and is parallel to line $r=i^+j^ +2k^+λ(i^−2j^ +3k^)$ meets the xy-plane.

Under which one of the following condition will the two planes x+y+z=7 andαx+βy+γz=3, be parallel (but not coincident)?

Distance of the point $P(p )$ from the line $r=a+λb$ is a. $∣∣ (a−p )+∣∣ b∣∣ _{2}((p −a)b˙)b ∣∣ $ b. $∣∣ (b−p )+∣∣ b∣∣ _{2}((p −a)b˙)b ∣∣ $ c. $∣∣ (a−p )+∣∣ b∣∣ _{2}((p −b)b˙)b ∣∣ $ d. none of these

Prove that the plane $r=(i^+2j^ −k^)=3$ contains the line $r=i^+j^ +λ(2i^+j^ +4k^)˙$