Class 12

Math

3D Geometry

Three Dimensional Geometry

Find the acute angle between the following planes.$x+y−z=4$ and $x+2y+z=9$.

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If $α,β,andγ$ are the an gles which a directed line makes with the positive directions of the co-ordinates axes, then find the value of $sin_{2}α+sin_{2}β+sin_{2}γ˙$

Direction ratios of two lines are $a,b,cand1/bc,1/ca,1/ab˙$ Then the lines are ______.

Determine whether the following pair of lines intersect or not. (1) $r=i^−5j^ +λ(2i^+k^);r=2i^−j^ +μ(i^+j^ −k^)$ (2) $r=i^+j^ −k^+λ(3i^−j^ );r=4i^−k^+μ(2i^+3k^)$

Find the equation the plane which contain the line of intersection of the planes$ri^+2j^ +3k^˙ −4=0andr2i^+j^ −k^˙ +5=0$ and which is perpendicular to the plane $r(5i^+3j^ −6k^)+8=0$ .

The line joining the points $(−2,1,−8)and(a,b,c)$ is parallel to the line whose direction ratios are $6,2,and3.$ Find the values of $a,bandc$

Find the equation of a line which passes through the point $(1,1,1)$ and intersects the lines $2x−1 =3y−2 =4z−3 and1x+2 =2y−3 =4z+1 ˙$

Show that the plane $2x−2y+z+12=0$ touches the sphere $x_{2}+y_{2}+z_{2}−2x−4y+2z−3=0.$

The plane which passes through the point $(3,2,0)$ and the line $1x−3 =5y−6 =4z−4 $ is a. $x−y+z=1$ b. $x+y+z=5$ c. $x+2y−z=1$ d. $2x−y+z=5$