PARAMETRIC: x ( t ) = y ( t ) = z ( t ) = SYMMETRIC: (quotient involving x ) = (

Question

PARAMETRIC:

x(t)=

y(t)=

z(t)=

SYMMETRIC:

(quotient involving x ) =

(quotient involving y) =

(quotient involving z ) =

Find the parametric and symmetric equations for the line passing through the points P1 = (2, 2, 3) and P2 = (1, 3, ?1). this problem does not have a unique answer unless we specify how to choose the direction vector v (see pages 610 and 611). In order to obtain the required solution for this problem use the vector v=P1?P2 (see page 585). Further, you need to use the given point P1 in the formula.

Explanation / Answer

parametric equations:

x = x0 + at

y = y0 + bt

z = z0 + ct

where (x0, y0, z0) is an initial point the line passes through

and <a, b, c> are the direction numbers of the line (vector)

P1 = (x0, y0, z0) = (2, 2, 3)

P1 - P2 = <a, b, c> = (2, 2, 3) - (1, 3, -1) = <1, -1, 4>

so parametric equations are:

x = 2 + t

y = 2 - t

z = 3 + 4t

symmetric equations:

(x - x0) / a = (y - y0) / b = (z - z0) / c

x: (x - 2) / 1 or x - 2

y: (y - 2) / (-1) or -y + 2

z: (z - 3) / 4

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