Vector equation of a line

You should already be familiar with the equation of a straight line in a 2D plane as being y = mx + c . This section takes that work a step further. It allows us to express the equation of a straight line in 3D space in a similar format.

Take a straight line through the point A with direction vector \mathbf{p} . R is another point on that line. The position vector (from an arbitrary origin) of A is \mathbf{a} , and of R is \mathbf{r} . The vector \overrightarrow{ \mathrm{AR} } is a multiple, t , of \mathbf{p} .

It follows that the position of a point R on the line is given by:
\mathbf{r} = \overrightarrow{ \mathrm{OR} } = \overrightarrow{ \mathrm{OA} } + \overrightarrow{ \mathrm{AR} } = \mathbf{a} + t \mathbf{p}

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2 July 2013, Created with GeoGebra