are given (a) Parametric form of vector equation Theorem . The vector equation of a straight line passing through a fixed point with position vector a and parallel to a given vector b is r tb , where t ∈ . Proof If a is the position vector of a given point A and r is the position vector of an arbitrary point P on the straight line, then AP . Since AP is parallel to b , we have = tb t ∈ ...
( ) or r = tb t ∈ ... ( ) This is the vector equation of the straight line in parametric form. Remark The position vector of any point on the line is taken as a tb (b) Non-parametric form of vector equation Since AP is parallel to b , we have AP b That is, ( This is known as the vector equation of the straight line in non-parametric form . (c) Cartesian equation Suppose P is ( , , ) x y z , A is ( , x y z and ˆ b i b j b k .
Then, substituting xi yj zk ˆ x i y j z k in ( ) and comparing the coefficients of ˆ ˆ , , i j k , we get tb y tb z tb ... ( ) Conventionally ( ) can be written as ... ( ) Fig. .
O P l a r Vector - - which are called the Cartesian equations or symmetric equations of a straight line passing through the point ( , x y z and parallel to a vector with direction ratios b b b . Remark (i) Every point on the line ( ) is of the form tb y tb z tb , where t ∈ . (ii) Since the direction cosines of a line are proportional to direction ratios of the line, if , l m n are the direction cosines of the line, then the Cartesian equations of the line are l m (iii) In ( ), if any one or two of b b b are zero, it does not mean that we are dividing by zero. But it means that the corresponding numerator is zero.
For instance, If , ≠ ≠ and b , then should be written as (iv) We know that the direction cosines of x - axis are , , . Therefore, the equations of x -axis are x − or , t y , where t ∈ . Similarly the equations of y -axis and z -axis are given by respectively.