P, Q, R and S are four points with the position vectors 3 i -4 j +5 k ,-4 i +5 j + k and -3 i +4 j +3 k , respectively. Then, the line P Q meets the line R S at the point

Step-by-step solution

Let the coordinates of four points P, Q, R and S be (3,-4,5), (0,0,4), (-4,5,1) and (-3,4,3) respectively. Now, equation of line P Q is aligned & x-3 0-3 = y+4 0+4 = z-5 4-5 & x-3 -3 = y+4 4 = z-5 -1 =r₁ aligned Equation of line R S is x+4 -3+4 = y-5 4-5 = z-1 3-1 x+4 1 = y-5 -1 = z-1 2 =r₂ Let (-3 r₁+3,4 r₁-4,-r₁+5 ) and (r₂-4,-r₂+5 . , .2 r₂+1 ) be the points on line (i) and (ii), respectively. Since, both lines intersect at a common point, then aligned & & -3 r₁+3 & =r₂-4 & & 3 r₁+r₂ & =7 and & & -r₂+5 & =4 r₁-4