NIELIT University(Govt. of India Institution) B.Tech/ M.Tech
ApplyLast Date to Apply : 15th September | Campuses in Agartala, Aizawl, Ajmer, Aurangabad, Calicut, Imphal, Itanagar, Kohima, Gorakhpur, Patna & Srinagar
Equations for a Line in Space is considered one of the most asked concept.
16 Questions around this concept.
The distance of the point (1, −5, 9) from the plane x − y + z = 5 measured along the
line x = y = z is :
A symmetrical form of the line of intersection of the planes is:
Equations for a Line in Space
A line is uniquely determined if
It passes through a given point and has given direction, or
It passes through two given points.
Equation of a line through a given point and parallel to a given vector
Let L be a line in space passing through point P(x0 , y0 , z0). Let be a vector parallel to L. Then, for any point on line Q(x, y, z), we know that vector PQ is parallel to vector b. Thus, there is a scalar, λ, such that
, which gives,
Cartesian Form
Vector equation of a line shows that the following equations are simultaneously true:
If we solve each of these equations for the component variables x, y, and z, we get a set of equations in which each variable is defined in terms of the parameter λ and that, together, describe the line. This set of three equations forms a set of parametric equations of a line:
If we solve each of the equations for λ assuming a, b, and c are non-zero, we get a different description of the same line:
These are parametric equations of the line. Eliminating the parameter λ from above equation, we get,
This is the Cartesian equation of the line.
NOTE:
If l, m, n are the direction cosines of the line, the equation of the line is
Equation of a line passing through two given points
Let P(x0 , y0 , z0) and Q(x1, y1, z1) be the points on a line, and let
Let be the position vector of an arbitrary point R(x, y, z) lying on this line
We want to find a vector equation for the line PQ.
Using P as our known point on the line, and as the direction vector, equation
gives
Cartesian Form
We can also find parametric equations for the line segment
As position vector of point R(x, y, z) is .
"Stay in the loop. Receive exam news, study resources, and expert advice!"