Class 12

Math

3D Geometry

Three Dimensional Geometry

Find the distance between the parallel planes $2x+3y+4z=4$ and $4x+6y+8z=12$.

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What is the distance between the planesx−2y+z−1=0 and−3x+6y−3z+2=0?

Find the angel between the following pair of lines: $r=2i^−5j^ +k^+λ(3i^+2j^ +6k^)andr=7i^−6k^+μ(i^+2j^ +2k^)$ $2x =2y =1z and4x−5 =1y−2 =8z−3 $

A line makes 45∘ with positive x-axis and makes equal angles with positive y, z axes, respectively. What is the sum of the three angles which the line makes with positive x, y and z axes?

The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point(0,172,−132). Then

If a line makes angles $α,βandγ$ with threew-dimensional coordinate axes, respectively, then find the value of $cos2α+cos2β+cos2γ˙$

If the center of the sphere ax2+by2+cz2−2x+4y+2z−3=0is (1/2,−1,−1/2), what is the value of b ?

Under what condition are the two linesy=mℓx+α,z=nℓx+β; and y=m′ℓ′x+α′,z=n′ℓ′x+β′ Orthogonal?

The plane $ax+by=0$ is rotated through an angle $α$ about its line of intersection with the plane $z=0.$ Show that he equation to the plane in the new position is $aby±za_{2}+b_{2} andα=0.$