Write the equation of a plan (in any form) through the point (4,3,7) and orthogonal the plane [x,y,z]= [2,4,-1]+ s[3,4,1]+t[-2,5,4]]

Question

Write the equation of a plan (in any form) through the point (4,3,7) and orthogonal the plane [x,y,z]= [2,4,-1]+ s[3,4,1]+t[-2,5,4]

Answer 1

i j k

3 4 1

-2 5 4

has determinant

11i - 14j + 23k

Then, the line connecting the 2 points has direction [4, 3, 7] - [2, 4, -1] = [2, -1, 8]

Then, the plane containing lines with directions [11, -14, 23] and [2, -1,8]

has equation

i j k

11 -14 23

2 -1 8

has determinant -89i - 42j + 17k

Then, consider the plane -89x - 42y + 17z = -363 (363 may be calculated by using the points on either plane)

We can also multiply through by -1 and have 89x + 42y - 17z = -363