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Application of Vectors to 3-Dimensional Geometry - A point on the straight line and the direction of the straight line | 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Chapter: 12th Mathematics : UNIT 6 : Applications of Vector Algebra

A point on the straight line and the direction of the straight line

(a) Parametric form of vector equation (b) Non-parametric form of vector equation

A point on the straight line and the direction of the straight line  are given

(A) Parametric form of vector equation

Theorem 6.11

The vector equation of a straight line passing through a fixed point with position vector  and parallel to a given vector  is  =  + t, where t R.

Proof

If  is the position vector of a given point and  is the position vector of an arbitrary point  on the straight line, then

 .


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  t.

(b) Non-parametric form of vector equation

Since  is parallel to  , we have   ×  

That is, ( − ) ×  = 0 .

This is known as the vector equation of the straight line in non-parametric form.

(c) Cartesian equation

Suppose is (xyz) , is (x1 , y1 , z1 ) and  b1 ˆi b2 ˆb3 ˆ. Then, substituting  x ˆy ˆz ˆ x1ˆy1ˆ j z1 ˆk  in (1) and comparing the coefficients of ˆ, ˆj, ˆ, we get

− x1 = tb1 , − y1 = tb2 , − z1 = tb3                     ………….(4)

Conventionally (4) can be written as


which are called the Cartesian equations or symmetric equations of a straight line passing through the point (x1y1 , z1) and parallel to a vector with direction ratios b1b2b3.

Remark

(i) Every point on the line (5) is of the form (x1 + tb1 , y1 + tb2 , z1 + tb3) , where ∈ R.

(ii) Since the direction cosines of a line are proportional to direction ratios of the line, if lmn are the direction cosines of the line, then the Cartesian equations of the line are


(iii) In (5), if any one or two of b1b2 , b3 are zero, it does not mean that we are dividing by zero. But it means that the corresponding numerator is zero. For instance, If b1 ≠ 0, b2 ≠ 0 and b3 = 0 , then 


(iv) We know that the direction cosines of x - axis are 1, 0,0 . Therefore, the equations of -axis are 


Similarly the equations of y -axis and z -axis are given by  respectively.


Tags : Application of Vectors to 3-Dimensional Geometry , 12th Mathematics : UNIT 6 : Applications of Vector Algebra
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12th Mathematics : UNIT 6 : Applications of Vector Algebra : A point on the straight line and the direction of the straight line | Application of Vectors to 3-Dimensional Geometry

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12th Mathematics : UNIT 6 : Applications of Vector Algebra


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