1 answer

How do I find the Cartesian equation of a plane containing two given lines?

Question:

How do I find the Cartesian equation of a plane containing two given lines?

Answers

Given two lines, they define a plane only if they are:
parallels non coincident or non coincident intersecting.

Explanation:

Given two lines, they define a plane only if they are:
parallels non coincident or non coincident intersecting. If they are parallels, taking a point in one of them and the support of the other we can define a plane. If they intersect, with the normal to both directions and their intersection point, a plane can also be constructed.

#1)# Parallel

#r_1->p = p_1 + lambda_1 vec v#
#r_2->p = p_2 + lambda_2 vec v#

#p_1^0 = p_1 + lambda_1^0 vec v#
#p_2^0 = p_2 + lambda_2^0 vec v#

#vec w = vec v xx (p_1^0-p_2^0) =vec v xx (p_1-p_2)#

the plane equation is given by

#Pi_1-><< vec w, (p-p_1)>> = << vec w, (p-p_2)>> = 0#

here #p = {x,y,z}#

#2)# Intersecting

Solve for the intersection point #p_0#

#p_0 = p_1 + lambda_1 vec v_1 = p_2 + lambda_2 vec v_2#

Once obtained #p_0# the plane is built as follows

#vec w = vec v_1 xx vec v_2#

#Pi_2 -> << vec w, (p-p_0) >> = 0#

.

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