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Vector and Cartesian Equations of a Line
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.
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
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
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)
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.
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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