Find the vector and Cartesian equations of the line through the point (1, 2, 4) and perpendicular to the two lines. ` vec r=(8 hat i-19 hat j+10 hat k)+lambda(3 hat i-16 hat j+7 hat k)` and ` vec r=(15 hat i+29 hat j+5 hat k)+mu(3 hat i+8 hat j-5 hat k)dot`

Find the vector and cartesian equations of the line through the point (1,2,-4) and perpendicular to the two lines- vec r=(8hat i-19hat j+10hat k)+lambda(3hat i-16hat j+7hat k) and vec r=(15hat i+29hat j+5hat k)+mu(3hat i+8hat j-5hat k)

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 and cartesan equation of the plane passing through the poin (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)) and vec(r) = hat(i) - 3 hat(j) + 5 hat(k) + mu (hat(i) + hat(j) - 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

If vec r=(hat i+2hat j+3hat k)+lambda(hat i-hat j+hat k) and vec r=(hat i+2hat j+3hat k)+mu(hat i+hat j-hat k) are bisector of two lines is

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

Find the equation in vector and cartesian form of the line passing through the point : (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))

(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)) .

vec r=(4hat i-hat j)+lambda(hat i+2hat j-3hat k) and vec r=(hat i-hat j+2hat k)+mu(2hat i+4hat j-5hat k)