Derive the equation of the line in space passing through a point and parallel to a vector both in vector and Cartesian form.
Open in App
Solution
A line is determined by a point and a direction. Thus, to find an equation representing a line in three dimensions choose a point P0 on the line and a non-zero vector ′v′ parallel to the line.
Since any constant multiple of a vector still points in the same direction, it seems reasonable that a point on the line can be found by starting at the point P0 on the line and following a constant multiple of the vector v (see the figure above).
If ′r′ is the position vector of P, then the line must be in the form
→r=→r0+t→v
Let the direction ratios of the line be a,b,c. Consider coordinates of any point A be (x,y,z).