Let L be the line given by `vec r = 2 hat i -2 hat j - hat k + lambda (- hat i + hat k)` and Let P be the point (2,-1,1). Also supposed that E be the plane containing three non collinear points `A(0,1,1); B(1,2,2) and C(1,0,1)`

Let L_1 be the line vec r_1=2 hat i+ hat j- hat k+lambda(hat i+2 hat k) and let L_2 be the line vec r_2=3 hat i+ hat j+mu(hat i+ hat j- hat k) . Let pi be the plane which contains the line L_1 and is parallel to L_2dot The distance of the plane pi from the origin is a. sqrt(6) b. 1//7 c. sqrt(2//7) d. none of these

Let L_1 be the line vec r_1=2 hat i+ hat j- hat k+lambda(hat i+2 hat k) and let L_2 be the line vec r_2=3 hat i+ hat j+mu(hat i+ hat j- hat k) . Let pi be the plane which contains the line L_1 and is parallel to L_2dot The distance of the plane pi from the origin is a. sqrt(6) b. 1//7 c. sqrt(2//7) d. none of these

Let L_1 be the line vec r_1=2 hat i+ hat j- hat k+lambda(i+2 hat k) and let L_2 be the line vec r_2=3 hat i+ hat j+mu(i+ hat j- hat k) . Let pi be the plane which contains the line L_1 and is parallel to L_2dot The distance of the plane pi from the origin is a. sqrt(6) b. 1//7 c. sqrt(2//7) d. none of these

Let L_1 be the line vec r_1=2 hat i+ hat j- hat k+lambda(i+2 hat k) and let L_2 be the line vec r_2=3 hat i+ hat j+mu(i+ hat j- hat k) . Let pi be the plane which contains the line L_1 and is parallel to L_2dot The distance of the plane pi from the origin is a. sqrt(6) b. 1//7 c. sqrt(2//7) d. none of these

Find vec a .(vec b × vec c) if vec a = 2 hat i+1 hat j+3 hat k and vec b=-1 hat i+2 hat j + hat k and vec c = 3 hat i + hat j +2 hat k

Find the vector equation of the plane that contains the line vec(r) = (hat(i) + hat(j) ) + lambda (hat(i) + 2 hat(j) - hat(k)) and the point (-1, 3,-4). Also , find the length of the perpendicular from the point (2,1,4) to the plane, thus botained.

The shortest distance between the line L_1=(hat i - hat j hat k) lambda (2 hat i - 14 hat j 5 hat k) and L_2=(hat j hat k) mu (-2 hat i - 4 hat 7 hat) then L_1 and L_2 is

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)) .

If line vec r=( hat i-2 hat j- hat k)+lambda(2 hat i+ hat j+2 hat k) is parallel to the plane vec r.(3 hat i-m hat k)=14 , then the value of m is (1). 2 (2) -2 (3) 0 (4) 3

The image of the point (1,2,3) about the line vec r=(2hat i+3hat j-hat k)+lambda(hat i2hat j+2hat k) is