How do you find an equation for the plane that contains the line with parametric equations $l=(8 - 7t, -5 - 2t , 5 - t)$ and is parallel to the line with parametric equations $x=3 + t, y=-7 + 9t, z=8 - 6t$ ?

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Answer 1 · 2016-09-30T10:56:43

$Pi->21 x - 43 y - 61 z-78=0$

Given two lines $L_1,L_2$ determine a plane $Pi$ such that

$L_1 in Pi$ and $L_2 !in Pi$

Here

$L_1 -> p = p_1 + lambda vec v_1$
$L_2 -> p = p_2 + lambda vec v_2$

with

$p_1 = (8,-5,5), vec v_1 = (-7,-2,-1)$
$p_2 = (3,-7,8), vec v_2 = (1,9,-6)$

The plane equation is

$Pi->p = p_1+lambda_1 vec v_1+lambda_2 vec v_2$ because

$L_1 in Pi$ ( make $lambda_2=0$ ) and is parallel to $L_2$ ( making $lambda_2=0$ )

The nonparametric plane equation can be easily obtained remembering that for the plane

$<< p-p_1,vec n >> = 0$

here $p = (x,y,z)$ and $vec n = vec v_1 xx vec v_2 = (21,-43,-61)$ so

$Pi->21 x - 43 y - 61 z-78=0$

How do you find an equation for the plane that contains the line with parametric equations l=(8 - 7t, -5 - 2t , 5 - t)l=(8 - 7t, -5 - 2t , 5 - t) and is parallel to the line with parametric equations x=3 + t, y=-7 + 9t, z=8 - 6tx=3 + t, y=-7 + 9t, z=8 - 6t?