Question

What is the symmetric equation of a line in three dimensional space?

Answer 1

Hint: From the question we have been asked to find the equation of a line in three dimensional space. For solving this question we will use the concept of three dimensional geometry. We will use the formulae of symmetric equation of a line with the direction vector passing through a point which is \[ \dfrac{x-{{x}_{0}}}{a}=\dfrac{y-{{y}_{0}}}{b}=\dfrac{z-{{z}_{0}}}{c}\]. Using this we will explain some examples and solve this question briefly. So, our solution will be as follows.

Complete step by step solution:
Generally in geometry that is in three dimensional geometry, the formulae of symmetric equation of a line with the direction vector \[=\left( a,b,c \right)\] passing through a point \[\left( {{x}_{0}},{{y}_{0}},{{z}_{0}} \right)\] will be as follows.
\[\Rightarrow \dfrac{x-{{x}_{0}}}{a}=\dfrac{y-{{y}_{0}}}{b}=\dfrac{z-{{z}_{0}}}{c}\]
Here the directional vector points can’t be zero, that is \[a,b,c\] can’t be zero.
If one of \[a,b,c\] is zero; for example, \[c=0\], then we can write as follows:
\[\Rightarrow \dfrac{x-{{x}_{0}}}{a}=\dfrac{y-{{y}_{0}}}{b}\] and \[z={{z}_{0}}\].
If two of \[a,b,c\] are zero; for example, \[b=c=0\], then we can write as follows.
\[y={{y}_{0}},z={{z}_{0}}\]
Here there is no restriction on x it can be any value that is it can be any real number.

Note: Students must be very careful in doing the calculations. Students must know the concept of three dimensional geometry very well to solve this question. We should know the formulae \[ \dfrac{x-{{x}_{0}}}{a}=\dfrac{y-{{y}_{0}}}{b}=\dfrac{z-{{z}_{0}}}{c}\] and the various conditions to solve this question briefly.