Three Dimensional Geometry
Find the angle between the lines x−3y−4=0,4y−z+5=0andx+3y−11=0,2y=z+6=0.
Let A(a⃗ ) and B(b⃗ ) be points on two skew line r⃗ =a⃗ +λ⃗ and r⃗ =b⃗ +uq⃗ and the shortest distance between the skew line is 1, where p⃗ and q⃗ are unit vectors forming adjacent sides of a parallelogram enclosing an area of 12units. If an angle between AB and the line of shortest distance is 60∘, then AB=
Find the equation of the plane passing through the point (−1,3,2) and perpendicular to each of the planes x+2y+3z=5and3x+3y+z=0.
Find the equation of the line passing through the intersection of 2x−1=3y−2=4z−3and5x−4=2y−1=z˙ and also through the point (2,1,−2)˙
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
The extremities of a diameter of a sphere lie on the positive y- and positive z-axes at distance 2 and 4, respectively. Show that the sphere passes through the origin and find the radius of the sphere.