Equation Of A Line In Three Dimensions - Practice Questions & MCQ

Equations for a Line in Space

A line is uniquely determined if 

1. It passes through a given point and has given direction, or 

2. 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 .