Equation of a line is defined as y= mx+c, where c is the y-intercept and m is the slope. Vectors can be defined as a quantity possessing both direction and magnitude. Position vectors simply denote the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin. 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
Since the line segment,
Also,
Therefore, α
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:
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):
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,
<
Eliminating α we have:
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
Point R lies on the line AB if and only if the vectors
Thus R lies on AB only if;
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:
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)
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
Eliminating α we have:
This gives us the Cartesian equation of a line.
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