Question

Provide two sets of geometric conditions that can be used to determine a line.

Short answer

To write the equation of a line in three - dimensional space we need a point on the line and a parallel vector on the line.

Step-by-step solution

Step 1:Given information

Geometric conditions that can be used to determine a line

Step 2:Explaination

A line in a plane can be determined by two distinct points or a single point and a direction, specified by a vector.

In $3d-$space lines cannot be described by a single linear equation, they are described by parametric equations.

A direction vector, which designates the direction of the line, and a point that passes through, we can obtain the equation of a line in a$3d-$space.

To write the equation of a line in $3d-$space we need a point on the line and a parallel vector on the line.

For the point }$\left(x_0,y_0,z_0\right)$and the direction vector }$(a,b,c)$ the line equation is as follows,

$r\left(t\right)=\left(x_0,y_0,z_0\right)+t(a,b,c)$