Equation of Line in Three Dimensions | Cartesian & vector Equation | Class 12 Math Notes Study Material Download Free PDF

Further, we shall study in detail about vectors and Cartesian equation of a line in three dimensions. It is known that we can uniquely determine a line if:

  • It passes through a particular point in a specific direction, or
  • It passes through two unique points

Let us study each case separately and try to determine the equation of a line in both the given cases.

Equation of a Line passing through a point and parallel to a vector

Let us consider that the position vector of the given point be  

\(\begin{array}{l}\vec{a} \end{array} \)  with respect to the origin. The line passing through point A is given by l and it is parallel to the vector \(\begin{array}{l}\vec{k} \end{array} \) as shown below. Let us choose any random point R on the line l and its position vector with respect to origin of the rectangular co-ordinate system is given by \(\begin{array}{l}\vec{r} \end{array} \).

Equation of Line

Since the line segment, \(\begin{array}{l}\overline{AR} \end{array} \)  is parallel to vector \(\begin{array}{l}\vec{k} \end{array} \) , therefore for any real number α, \(\begin{array}{l}\overline{AR} \end{array} \) = α  \(\begin{array}{l}\vec{k} \end{array} \)

Also, 

\(\begin{array}{l}\overline{AR} \end{array} \) = \(\begin{array}{l}\overline{OR} \end{array} \) –  \(\begin{array}{l}\overline{OA} \end{array} \)

Therefore, α \(\begin{array}{l}\vec{r} \end{array} \) = \(\begin{array}{l}\vec{r} \end{array} \) – \(\begin{array}{l}\vec{a} \end{array} \)

From the above equation it can be seen that for different values of α, the above equations give the position of any arbitrary point R lying on the line passing through point A and parallel to vector k. Therefore, the vector equation of a line passing through a given point and parallel to a given vector is given by:

\(\begin{array}{l}\vec{r} \end{array} \) = \(\begin{array}{l}\vec{a} \end{array} \) + α \(\begin{array}{l}\vec{k} \end{array} \)

If the three-dimensional co-ordinates of the point ‘A’ are given as (x1, y1, z1) and the direction cosines of this point is given as a, b, c then considering the rectangular co-ordinates of point R as (x, y, z):

3d vector

Substituting these values in the vector equation of a line passing through a given point and parallel to a given vector and equating the coefficients of unit vectors i, j and k, we have,

3d 2

Eliminating α we have:

3d 3

This gives us the Cartesian equation of line.

Equation of a Line passing through two given points

Let us consider that the position vector of the two given points A and B be \(\begin{array}{l}\vec{a} \end{array} \) and \(\begin{array}{l}\vec{b} \end{array} \)  with respect to the origin. Let us choose any random point R on the line and its position vector with respect to origin of the rectangular co-ordinate system is given by \(\begin{array}{l}\vec{r} \end{array} \).

Equation of a Line

Point R lies on the line AB if and only if the vectors \(\begin{array}{l}\overline{AR} \end{array} \) and \(\begin{array}{l}\overline{AB} \end{array} \)  are collinear. Also, \(\begin{array}{l}\overline{AR} \end{array} \) =  \(\begin{array}{l}\vec{r} \end{array} \) – \(\begin{array}{l}\vec{a}\end{array} \) \(\begin{array}{l}\overline{AB} \end{array} \) =  \(\begin{array}{l}\vec{b} \end{array} \) – \(\begin{array}{l}\vec{a}\end{array} \)

Thus R lies on AB only if;

\(\begin{array}{l}\vec{r} – \vec{a} = \alpha (\vec{b} – \vec{a})\end{array} \)

Here α is any real number.
From the above equation it can be seen that for different values of α, the above equation gives the position of any arbitrary point R lying on the line passing through point A and B. Therefore, the vector equation of a line passing through two given points is given by:

\(\begin{array}{l}\vec{r} = \vec{a} + \alpha (\vec{b} – \vec{a})\end{array} \)

If the three-dimensional coordinates of the points A and B are given as (x1, y1, z1) and (x2, y2, z2) then considering the rectangular co-ordinates of point R as (x, y, z)

Equation of Line

Substituting these values in the vector equation of a line passing through two given points and equating the coefficients of unit vectors i, j and k, we have

Equation of Line

Eliminating α we have:

Equation of Line

This gives us the Cartesian equation of a line.

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.