(a) Parametric form of vector equation (b) Non-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*âˆˆ R.

Proof

If * *is the position vector of a given point *A *and is the position vector of an arbitrary point *P *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*.

Since * *is parallel to * *, we have * *Ã— * *=

That is, (* *âˆ’ ) Ã— * *= 0 .

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

Suppose *P *is (*x*, *y*, *z*) , *A *is (*x*1 , *y*1 , *z*1 ) and * *= *b*1 Ë†*i* + *b*2 Ë†*j *+ *b*3 Ë†*k *. Then, substituting * *= *x* Ë†*i *+ *y* Ë†*j *+ *z* Ë†*k *, * *= *x*1Ë†*i *+ *y*1Ë†* j *+ *z*1 Ë†*k* in (1) and comparing the coefficients of Ë†*i *, Ë†*j*, Ë†*k *, we get

*x *âˆ’ *x*1 = *tb*1 , *y *âˆ’ *y*1 = *tb*2 , *z *âˆ’ *z*1 = *tb*3 â€¦â€¦â€¦â€¦.(4)

Conventionally (4) can be written as

which are called the Cartesian equations or symmetric equations of a straight line passing through the point (*x*1, *y*1 , *z*1) and parallel to a vector with direction ratios *b*1, *b*2, *b*3.

(i) Every point on the line (5) is of the form (*x*1 + *tb*1 , *y*1 + *tb*2 , *z*1 + *tb*3) , where *t *âˆˆ R.

(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

(iii) In (5), if any one or two of *b*1, *b*2 , *b*3 are zero, it does not mean that we are dividing by zero. But it means that the corresponding numerator is zero. For instance, If *b*1 â‰ 0, *b*2 â‰ 0 and *b*3 = 0 , then

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