Find the equation of the plane passing through the point `(1, 1, 1)` and containing the line `vec r = (-3hat i + hat j+ 5hat k) + lambda(3hat i -hat j– 5hat k)` Also, show that the plane contains the lines `vec r = (-hat i +2hat j +5hat k) + lambda(hat i – 2hat j – 5hat k)`

Find the equation of the plane passing through the point P(1,1,1) and containing the line vec(r) = (-3 hat(i) + hat(j) + 5 hat(k)) + lambda (3 hat(i) - hat(j) - 5 hat(k)) . Also, show that the plane contains the line vec(r) = (- hat(i) + 2 hat(j) + 5hat(k)) + mu (hat(i) - 2 hat(j) - 5 hat(k)) .

Find the equation of the line passing through the point hat i+ hat j- hat k and perpendicular to the lines vec r= hat i+lambda(2 hat i+ hat j-3 hat k)a n d vec r=(2 hat i+ hat j- hat k)dot

Find the equation of the line passing through the point hat i+ hat j- hat k and perpendicular to the lines vec r= hat i+lambda(2 hat i+ hat j-3 hat k)a n d vec r=(2 hat i+ hat j- hat k)dot

Find the vector equation of the plane passing through three points with position vectors hat i+hat j-2hat k,hat i-hat j+hat k and hat i+2hat j+hat k .Also find the coordinates of the point of intersection of this plane and the line vec r=3hat i-hat j-hat k+lambda(2hat i-2hat j+hat k)

Find the vector equation of the plane passing through three points with position vectors hat i+hat j-2hat k2hat i-hat j+hat k and hat i+2hat j+hat k Also find the coordinates of the point of intersection of this plane and the line vec r=3hat i-hat j-hat k+lambda(2hat i-2hat j+hat k)

Find the shortest distance between the lines vec r = (hat i + 2 hat j + hat k) + lambda (hat i - hat j + hat k) and vec r = ( 2 hat i - hat j - hat k) + mu ( 2 hat i + hat J + 2 hat k)

Find the angle between the line vec r=(2hat i-hat j+3hat k)+lambda(3hat i-hat j+2hat k) and the plane vec r*(hat i+hat j+hat k)=3

Find the vector and cartesian equation of a plane containing the two lines vec r=(2hat i+hat j-3hat k)+lambda(hat i+2hat j+5hat k) and vec r=(3hat i+3hat j+2hat k)+mu(3hat i-2hat j+5hat k) Also,show that the line vec r=(2hat i+5hat j+2hat k)+P(3hat i-2hat j+5hat k) lies in the plane.

Find the distance of the point (-1,\ -5,\ -10) , from the point of intersection of the line vec r=(2 hat i- hat j+2k)+lambda\ (3 hat i+4 hat j+2 hat k) and the plane vec rdot( hat i- hat j+ hat k)=5

The distance between the line vec r=(2 hat i-2 hat j+3 hat k)+lambda( hat i- hat j+4 hat k) and plane vec rdot( hat i+5 hat j+ hat k)=5.