Question

(a) Find the cosine of the angle between the lines L1 and L2 whose vector equations are given below:

L1 : ~r1(t) = [1, 1, 1] + t[1, 2, 3]

L2 : ~r2(t) = [1, 1, 1] + t[−1, 4, 2].

(b) Find the equation of the plane that contains both L1 and L2.

Answer 1

a). Let Ɵ be the angle between the lines L₁ and L₂. These 2 lines pass through the point (1,1,1) and are parallel to the vectors (1,2,3) and (-1,4,2) respectively.

Thus, Ɵ is the angle between the vectors u = (1,2,3) and v = (-1,4,2). Thus, cos = (u.v)/|u|.|v| = (-1+8+6)/[√(1+4+9)][√(1+16+4)] = 13/(√14)(√21) = 13/7√6.

b). Let n = (p,q,r) be a vector normal to the plane that contains the lines L₁ and L₂. Then the vector n = (p,q,r) is orthogonal to both the vectors (1,2,3) and (-1,4,2) . Hence (p,q,r).(1,2,3) = 0 or, p+2q+3r = 0…(1) . Also, (p,q,r).(-1,4,2) = 0 or,-p+4q+2r = 0…(2). On solving these equations, we get p = -4r/3 and q = -5r/6 so that n = (-4r/3,-5r/6,r) = -(r/6)( 8,5,-6) = t(8,5,-6) where t = -r/6 is an arbitrary real number.

Assuming t = 1, we have n = (8,5,-6) . Then the equation of the required plane is (8,5,-6) .( x-1,y-1,z-1) = 0 or, 8x-8+5y-5-6z+6 = 0 or, 8x+5y-6z = 7.