Introduction to Quaternions

32. EQUATIONS of a straight line.

1. Let β be a vector (unit or otherwise) parallel to or along the straight line; α the vector to a given point A in the line, ρ that to any point whatever P in the line, starting from the same origin O; then AP is a vector parallel to β

  = xβ, say,

  OP = OA + AP 

  ρ = α + xβ (1) 

2. Another form in which the equation of a straight line may be expressed is this: let OA = α, OB = β be the vectors to two given points in the line; then

  AB = β - α  and  AP = x(β - α);

∴ ρ = α + x(β - α) (2). 

Of course the β of No. 2 is not that of No. 1. The first form of the equation supposes the direction of the line and the position of one point in it to be given, the second form supposes two points in it to be given.

3. A third form may be exhibited in which the perpendicular on the line from the origin is given.