Find the equation of the line passing through the point (1, –1, 1) and perpendicular to the lines joining the points (4,3,2), (1,–1,0) and (1,2,–1), (2, 1, 1).
The Cartesian equation of a line passing through a point (x1, y1, z1) and having directional ratios proportional to a,b,c is given by,
The required line passes through the point (1,–1,1), now we need to find the direction ratios of the line which are a,b,c . this equation of the required line is,
Direction ratios of the line joining A(4,3,2) and B(1,–1,0)
= (4–1,3+1,2–0) = (3,4,2)
Direction ratios are 3,4,2
Direction ratios of the line joining C(1,2,–1) and D(2,1,1)
= (2–1,1–2,1+1) = (1,–1,2)
Direction ratios are 1,–1,2
It is given that the line AB is perpendicular to the required line, so the dot product equation will be equal to zero .
a×3 + b×4 + c×2 = 0
3a+4b+2c = 0 ……….(i).
It is given that line CD is perpendicular to the required line, so the dot product will be equal to zero .
a×1 + b×(–1) + c×2 = 0
a–b+2c = 0 ………(ii).
Solving equations (i) and (ii) by cross multiplication method, we get
= λ
a = 10λ, b = –4λ, c = –7λ
Therefore, the cartesian or symmetry form of equation of the required line is,
