Find the equation of a line passing through a point (2, -1, 3) and parallel to the line ` -> r=( hat i+ hat j)+lambda(2 hat i+ hat j-2 hat k)dot`

Find the vector equation of the line passing through (2,1,-1) and parallel to the line vec r=(hat i+hat j)+lambda(2hat i-hat j+hat k)* Also,find the distance between these two lines.

Find the vector and Cartesian forms of the equation of the plane passing through the point (1,\ 2,\ -4) and parallel to the lines vec r=( hat i+2 hat j-4 hat k)+lambda(2 hat i+3 hat j+6 hat k)a n d\ vec r=( hat i-3 hat j+5 hat k)+mu( hat i+ hat j- hat k)dot Also find the distance of the point (9, -8, -10) from the plane thus obtained.

Find the vector equation of a line passing through the point (2,3,2) and parallel to the line : vec(r) = (-2 hat(i) + 3 hat(j) ) + lambda (2 hat(i) - 3 hat(j) + 6 hat(k)) . Also, find the distnace between these two lines.

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

(A ) Find the vector and cartesian equations of the line through the point (5,2,-4) and which is parallel to the vector 3 hat(i) + 2 hat(j) - 8 hat(k) . (b) Find the equation of a line passing through the point P(2, -1, 3) and perpendicular to the lines : vec(r) = (hat(i) + hat(j) - hat(k) ) + lambda (2 hat(i) - 2 hat(j) + hat(k)) and vec(r) = (2 hat(i) - hat(j) - 3 hat(k) ) + mu (hat(i) + 2 hat(j) + 2 hat(k)) .

Find the equations of a line passing through the point P(2,-1,3) and perpendicular to the lines vec(r ) =(hat(i) + hat(j) -hat(k)) +lambda (2hat(i) -2hat(j) +hat(k)) and vec( r) =(2hat(i) -hat(j) -3hat(k)) +mu (hat(i) +2hat(j) +2hat(k))

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 (1,1,1) and containing the line vec r=(-3hat i+hat j+5hat k)+lambda(3hat i-hat j5hat k) Also,show that the plane contains the lines vec r=(-hat i+2hat j+5hat k)+lambda(hat i2hat j5hat k)

Find the equation of lane passing through the point hat i+ hat j+ hat k and parallel to the point -> rdot(2 hat i- hat j+2 hat k)=5.