. . Straight Line passing through two given points (a) Parametric form of vector equation Theorem . The parametric form of vector equation of a line passing through two given points whose position vectors are a and b respectively is ), t b a t ∈ .
(b) Non-parametric form of vector equation The above equation can be written equivalently in non-parametric form of vector equation as ) ( (c) Cartesian form of equation Suppose P is ( , , ) x y z , A is ( , x y z and B is . Then substituting xi yj zk ˆ x i y j z k ˆ x i y j z k in theorem . comparing the coefficients of ˆ ˆ , , i j k , we get ), ), t x t y t z and so the Cartesian equations of a line passing through two given points ( , x y z are given by Fig. .
O P B a r ( , x y z ( , , ) x y z y z Vector - - Applications of Vector Algebra From the above equation, we observe that the direction ratios of a line passing through two given points ) ( , x y z and are given by x y y z , which are also given by any three numbers proportional to them and in particular Example . A straight line passes through the point ( , , ) and parallel to . Find (i) vector equation in parametric form (ii) vector equation in non-parametric form (iii) Cartesian equations of the straight line. The required line passes through ( , , ) .
So, the position vector of the point is Let = + . Then, we have (i) vector equation of the required straight line in parametric form is tb t ∈ . Therefore, ) ( ), ∈ . (ii) vector equation of the required straight line in non-parametric form is ( Therefore, )) ( ) (iii)