AHL 3.14 (HL)—Vector equations of lines

A vector line is a point plus a direction.

The equation r=a+λd generates every point reached from a known point a by scaling the direction vector d.

Example

The line through (1,2,0) parallel to (2,−1,3) is r=(1,2,0)+λ(2,−1,3); λ=2 gives (5,0,6).

Match coordinates to solve for a parameter, then compare direction vectors to test parallelism or intersection.

Different equations can describe the same line; a shared point alone does not prove two lines intersect unless the parameters agree.

Equivalent forms for a line through $(x_0,y_0,z_0)$ with direction $(l,m,n)$ are $\mathbf r=\mathbf a+\lambda\mathbf d$, $x=x_0+\lambda l$, $y=y_0+\lambda m$, $z=z_0+\lambda n$, and $(x-x_0)/l=(y-y_0)/m=(z-z_0)/n$ where the displayed denominators are non-zero. For two lines, use their direction vectors in the dot-product formula for the acute angle. In a motion model, $\lambda$ may represent time, $\mathbf d$ velocity and $|\mathbf d|$ speed.