SOLUTION: Find the vector equation for the line of intersection between r.(4i + 3j + 7k) = 12 and r.(2i + j + 3k) = 4

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Question 1114443: Find the vector equation for the line of intersection between r.(4i + 3j + 7k) = 12 and r.(2i + j + 3k) = 4
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The normals for the plane are (4,3,7) and (2,1,3).
Find the cross product of these two vectors, it's parallel to the line of intersection,
axb=%28matrix%283%2C3%2Ci%2Cj%2Ck%2C%0D%0A4%2C3%2C7%2C2%2C1%2C3%29%29
axb=%283%2A3-1%2A7%29i%2B%287%2A2-3%2A4%29j%2B%284%2A1-2%2A3%29k
axb=%289-7%29i%2B%2814-12%29j%2B%284-6%29k
axb=%289-7%29i%2B%2814-12%29j%2B%284-6%29k
axb=%282%29i%2B%282%29j%2B%28-2%29k
(2,2,-2)
Solve for any point on the line using the equations.
You can set z=0 and solve for x and y of a point,
4x%2B3y=12
2x%2By=4
leads to,
4x%2B3y-4x-2y=12-8
y=4
So then,
2x%2B4=4
x=0
So (0,4,0) is one point on the line and with the results from above,
x=0%2B2t
y=4%2B2t
z=0-2t
.
.
x=2t
y=4%2B2t
z=-2t