12.5 Equations of lines and planes.pdf - in RZ is given by A line direction point 1123 slope a point on the line and a form need point Po Yo yo Zo on a

# 12.5 Equations of lines and planes.pdf - in RZ is given by...

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A line in RZ is given by a point on the line and a direction point slope form Likewise in 1123 we need a point Po Yo yo Zo on a line L and a direction of L In 1123 the direction of a line is conveniently by a vector that is parallel to L Let T be a vector parallel to L Let PIX Y H be an arbitrary point ou L Let rT and T be the position vectors of Po ie p Denote them Tpo and OT I is the vector with representation pTp Then F root I
But since a and J are parallel then there is a scalar t s t a tv thus I This is the vector equation of L Each value of the parameter t gives the position vector f of a point on L As t varies we trace the line L by the tip of T J L a b c Recall that it gives the direction to Cta tb to F L X Y Z to L Yo Yo 707 Then our vector eq 1h F root tv becomes 4h47 L Xo Yo Zo t Lta tb to LX Y z Liotta Yo ttb 2 otto Look at the components They are the parametric equations of the line L through Po Ho yo fo
and parallel to T La b c Each value off gives a point 44471 on our line Parametric eq us for a line through Ko Yo Za and parallel to La b c are x Xotaty yotbtz zot find a vector edu and parametric egths for the line

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