Vector and Cartesian Equations of a Line

Created: Thursday, 15 September 2011 09:48 | Published: Thursday, 15 September 2011 09:48 | Written by Super User | Print

Equation of a line in space

We have studied equation of lines in previous classes. Now we will learn the vector and Cartesian equation of a line in space.

v_equation_1
A line is uniquely determined if,
i) It passes through a given point and has given direction
ii) It passes through two given points.

Equation of a line through a given point and parallel to a given vector

Let a be given position vector of the given point and b be the given vector, then its equation is given by r=a+λb

v_equation_2

Vector Equation

In the above figure AP is parallel to b, so AP=λb ………(1)

AP= OP – OA

   = r – a

(1)    becomes,     r – a = λb

                 r= a+λb, which is the vector equation.

Hence vector equation of a line passing through a point with position vector a and parallel to a given vector b is given by r=a+λb

Cartesian Equation

Let the coordinates of the given point be A(x₁, y₁, z₁) and the direction ratios of the parallel vector be <a, b, c>. Let P(x, y,z) be any point (General point) on the line.

The Cartesian equation is given by

Example: Find the vector and Cartesian equations of the line through the point (5,3,-5) and which is parallel to the vector 4î-7ĵ+3k

Solution: We have a=5î+3ĵ-5k and b= 4î-7ĵ+3k, so

Vector equation is r= a +λb

= (5î+3ĵ-5k) +λ (4î-7ĵ+3k)

Cartesian Equation is

Equation of a line passing through two points

Let a and b be the position vectors of two points that are lying on a a given line then their equation is given by r= a+λ(b-a)

v_equation_8

Vector Equation

Let a and b be the position vectors of the points lying on the line and r be the position of any point (general point).

We know AP and AB are collinear vectors, therefore P will lie on the line if and only if AP = λ AB

r-a = λ (b-a)

r = a +λ (b-a), which is the vector equation.

Cartesian Equation

Let A(x1, y1, z1) and B(x2, y2, z2) be two point in the line and P(x, y, z) be a general point on the line, the Cartesian Equation is given by



Example: Find the Vector and Cartesian equation of the line joining the points (-1,3,2) and (3,0,1)

Solution: Here a= -î+3ĵ+2k and b=3î+0ĵ+k

            Vector equation is r= (-î+3j+2k)+λ(4î-3ĵ-k)

Cartesian equation is

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